Here are my thoughts on cycling/cycled/cycle based almost entirely on researching online and reading saltwater reference books with very little personal actual experience or experimentation.
Some terminology distinctions (term is italicized when used in text):
"Bioload" is being used as a very general term to indicate the total contents of living organisms and nutrient introduction/conversion in the tank. This includes animals that convert nutrients to ammonia, feeding rate or nutrients added within food or other additives, consumption rates of nutrients by different organisms, etc.
"Cycle" means the conversion of one ammonia molecule to one nitrite molecule to one nitrate molecule. Each cycle is the conversion of only one molecule, as I am using the terminology here.
"Cycler(s)" means one unit of nitrifying bacteria or other nitrifying organisms, or the biological capacity to complete one cycle.
"Cycled" to me is a nearly worthless term, as it just indicates that at least one cycle has been completed with no indication of cycling capacity. I realize this goes against current terminology, but I consider it too vague a term to be useful.
"Cycling" means that there are enough cyclers for the current tank bioload. This is independent of whether "bioload" is being supplied by fish/invertebrates/corals or by measured additions of chemical nutrients, primarily ammonia for initial cycles.
"Cycling capacity" is the number of cycles completed in some unit of time which is entirely dependent on number of cyclers and is ultimately the measure of capability of the system to handle bioload.
Ideally, a system should operate with a balanced cycling capacity and bioload which is indicated by undetectable nutrients due to enough cyclers to complete cycles very shortly after the introduction of even tiny amounts of ammonia/nitrites/nitrates. Under proper conditions, the cycles happen so fast that the only detectable nutrient is nitrates (unless there are also enough anaerobic bacteria to convert nitrates to nitrogen gas and expel it from the tank). To truly complete this idea of a system under ideal operation parameters, macroalgaes or other algaes could be used so that even nitrate and phosphate are kept quite low, but this goes beyond the discussion about nitrification cycling.
Breaking my response down based on poll possible responses:
"A cycle is only complete when Ammonia is "0", Nitrite is "0" and Nitrates are rising"
I think this is not true, as a cycle is technically complete once any ammonia is converted to nitrite that is then converted to nitrate, so presence of nitrate is an indication of at least one completed cycle. This is not to say that one cycle is an indication that there are enough cyclers to complete the loop often enough to provide enough cycling capacity for a higher bioload.
"A. The best way to cycle is to add an ammonia source alone and 'wait' for the steps above to finish"
Adding an ammonia source and waiting seems to be an effective method by harvesting free nitrifying bacteria cyclers, but it seems to be slower than adding an initial boost of cyclers. This may help teach the patience needed for marine aquariums, so it is not all bad to just wait.
"B. The best way to cycle is to add ammonia, and bacteria and 'wait' for the steps above to finish"
This seems to be the second-best method of starting a tank to reach an initial cycling tank with a certain cycling capacity (first-best is introduction of a significant amount of live rock from the ocean or an established system, especially with good growth of organisms on it). This seems to be experimentally proven to be able to occur pretty quickly. Of course, the cycling capacity needs to be considered when planning how heavy the bioload will be and should be measured through indicators like ammonia decrease in a certain time period.
"C. The best way to cycle a tank is to add bacteria and fish on day 1 (per instructions)"
Though this method works and has been done for years, I am undecided on this method. Though I think there are some benefits to doing this more natural method, I also do not like the direct intentional stress on the first fish (even if they are "hardy"). The chemical method using compounds such as ammonia chloride seems more humane but may leave out important biological details that are just not yet understood fully.
"A. You can take all of the stuff (rock, filter, fish, coral) and put in a new tank"
If by this we consider a total tank transfer, I think this is very doable and has been proven by many in the past. If the cycling capacity is decreased while bioload remains the same (i.e. only using part of the live rock or filter media while transferring all fish/corals), there may still be ammonia detectable for a time until the nitrifying bacteria cyclers increase and bring up the cycling capacity to match current bioload. Due to actual measurement methods, the spike in detectable ammonia and nitrites may be too small to see but is dependent on how much lower the cycling capacity is than current bioload.
"B. If you move rock, fish, coral to a new tank, you will have a cycle"
Well, by my definitions, yes, there will be a new cycle (as I consider new cycles to be occurring continually upon introduction of nutrients in the presence of any nitrifying cyclers). By what I consider the intention of this response, a measurable spike in ammonia and/or nitrites should only occur if bioload from transferred items/occupants is increased proportional to cycling capacity (or cycling capacity is decreased in proportion to bioload). If not, ammonia and/or nitrites should not increase above the cycling capacity to be converted and remain unmeasureable.
"A. If you add significant bioload to a 'cycled tank, you risk a new cycle"
Yes, I think so, since there will be a lag time of greater introduction than conversion until the nitrifying cyclers multiply enough to balance cycle capacity with bioload. This assumes that there are not a lot of dormant nitrifying cyclers that have not died off after previous excess multiplication. I don't know numbers (how much bioload increase, how long until nitrifying cyclers starve, etc.) on this, but I assume that there is potential for some increase in bioload to be manageable due to starving/dormant-but-not-dead-yet nitrifying cyclers. If bioload increases match the multiplication rate of cyclers, the cycling capacity may increase fast enough to balance biolad and keep ammonia/nitrites undetectable.
"B. If you add large bioload, to a cycled tank, nothing will happen - all the bact are there"
As stated in the previous option, this may be true in certain circumstances but is dependent on a lot of factors such as actual total volume, surface/appropriate conditions for nitrifying cyclers, amount of current bioload, actual amount of bioload increase, etc.
If I cycled a 100-gallon aquarium with one small rock and only one 2" damsel until cycling capacity balanced with bioload to ensure fast enough cycles occurring to keep ammonia and nitrite undetectable, then added ten 10" groupers and fed even sparsely but enough to keep them alive, it would seem guaranteed that ammonia will spike tremendously (and thus nitrite and nitrate will also spike as cycles complete). This obvious extreme demonstrates the principle behind my opinion here.
I think of nitrification like package delivery or production. There are 3 types of delivery vehicles (cyclers): A carries/converts ammonia to nitrite, B carries/converts nitrite to nitrate, and C carries/converts nitrate to nitrogen gas. For nitrification, we can ignore C and focus entirely on A and B.
If you start the delivery business small (few cyclers) because demand is small (low bioload, therefore low cycling capacity necessary), the few packages (nutrients) can be delivered with few cyclers. As demand increases (raising bioload), more packages need to be delivered (cycling capacity must increase). Since it takes time to hire more deliverers (cyclers, increased potentially with bottled bacteria or introduction of mature filters/media/live-rock/live-sand/biofilm/old water) and there are two different types of deliverers that need hired (A and B), the higher demand cannot be processed immediately and the warehouse (either A, B, or A and B) starts to fill with packages (nutrients). If demand increases faster (higher bioload than increase of cyclers/cycling capacity) than delivery (cycles), there will always be a detectable surplus of packages at the warehouse (ammonia/nitrite measureable in the tank/system). By hiring more deliverers (increase cyclers), deliveries (cycles) will occur faster to deal with the surplus of packages (nutrients) until a steady stream of deliveries keeps the surplus undetectable at the warehouse. At perfect balance, every package (ammonia/nitrite) is picked up immediately so there is never a surplus at the warehouse (nutrients not detectable) and there are no extra deliverers (cyclers). Eventually, there may be too many deliverers (cyclers) over-hired when there are not enough packages for each one to carry (bioload less than or balanced with cycling capacity). This results in deliverers quitting or being fired (eventual starvation/die-off of excess cyclers) until balance is again restored.
In our delivery system, there are two stages (A and B) that are different. The first deliverer type A carries a package that contains a bomb with potential to go off and will become more destructive the more packages there are in surplus at a given time (ammonia is quite acutely toxic), so it is extremely critical that warehouse A does not have a surplus of packages (ammonia) that could be damaging to nearby occupants (fish/etc.). At warehouse B, the bombs have been diffused but the contents remaining may still be somewhat harmful, maybe mostly just stressful, to carry (nitrite is non-toxic/not as toxic as ammonia/only a stress on marine fish, still an ongoing discussion with different viewpoints whether it needs to be measured), thus the deliveries from warehouse B are not as critical for the well-being of the nearby occupants but would still be a surplus at the warehouse if not delivered, thus indicating an incomplete cycle.
Once the package gets delivered by B (converted to nitrate), it is no longer dangerous but lures a lot of looters (undesirable bacteria/algae/organisms due to high nitrates) that can just cause problems if the packages are not moved to the final customer and removed from the delivery service (converted to nitrogen gas and expelled from the system).
Well, that is a lot of text, but I hope it helps this discussion and that I can continue to learn and better-understand the processes in these complex biological ecosystems we keep in aquariums!