Hopefully this helps put the P issue to bed. I have a reference below that hopefully will clarify things.
with regard to Total "P"... the Hanna meter will never give you total P and i believe that is acknowledged in the instructions. To get total P you can't shake and read. You have to "digest" the sample for hours at 300 degrees C to break down the organics. For example.. DNA is full of phosphorus.. but make a DNA sample in water and the hanna meter ain't gonna show that because "shake with no bake" is not gonna break DNA down into phosphate which is what the meter is looking for. The assumption with the hanna process is "most" of the P containing organics are broken down into PO4 during that shake.. but not all. For the purposes of a reef tank none of that probably matters and i don't see the need for doing phos by ICP when you have Hanna ULM unless you are doing research. The more complex P we don't read with the hanna is likely inconsequential.
Now that hopefully everyone sees that the hanna method cannot guarantee total P let's looks at ICP vs. Colorimetric (Hanna). In the real world where data counts samples must be digested for the colorimetric analysis of Total P to be effective. This digestions occurs in sulfuric acid at high temps for hours. In ICP.. your sample is aspirated into a plasma at 6,000 kelvin (temp of the sun). In theory ALL of the organic P is broken down at that temperature which is why ICP values should be greater than or equal to a hanna value. If your Hanna sample was "digested" prior to analysis then now you can make the argument the values should be equal between Hanna and ICP.
Here is the abstract for the reference below. Keep in mind colorimetric analysis is "digested" when comparing to ICP.
The possibility that P concentrations in extracts, digests, and water samples might be different when measured with inductively coupled plasma spectrometry (ICP) as compared to colorimetric procedures has been debated since the introduction of ICP instruments. The ability of ICP to measure several elements simultaneously has increased interest in multi-element extracts for routine soil testing and likely reduced the use of the colorimetric approach. However, colorimetric procedures offer some advantages, such as increased sensitivity and lower instrumentation costs, and it is unlikely that ICP will completely displace colorimetric procedures. The commonly held belief is that ICP would measure greater P concentrations than colorimetric procedures because the high temperature environment of the plasma would allow the measurement of organic P compounds or other soluble P complexes that would not be measured colorimetrically. Differences in P concentrations measured by these two methods may have implications for agronomic and environmental P testing. The purpose of this paper is to discuss potential differences in P determination by ICP versus colorimetric procedures.
Nice find on the paper...Good read!
So I will start with a quote from the Summary and Conclusions of this paper
"The method of P determination used in water samples and extracts of soils and by-products can influence the reported concentrations.
For most sample types, ICP is slightly greater than colorimetric analyses, but the two methods are well correlated for high P samples.
However, extreme caution needs to be used for samples with low P concentrations (e.g., 60 mg/kg extractable P), as the difference may be as great as 5 fold. "
This matches quite nicely with my understanding of what to expect, as you well point out above. I should expect that any measurement taken by a Hanna Instrument would not pick up total P. Where as the ICP testing would pick up total P. This would necessarily mean, as the paper you provided states, the ICP measurement will be slightly greater in total P than that of the Hanna Instrument. I seems according to the paper that as you move down in concentration the difference get greater.
This is exactly what I expected when I sent in my first ICP test and compared it to my Hanna results, but to my surprise the ICP reported a lower value for P and PO4....not by a small amount by close to 50% (My measurement 20 ppb P ....ICP 10 ppb P) I concluded that my measurement was incorrect so I went back to my retained sample of the test I sent for ICP analysis and retested....Sure enough I got a lower reading 12 ppb----close enough! So that should have been the end of the story, but it was not lucky!!.. I continued to send in ICP tests every 90 days (Started in 2017) and I saw exactly the same results...Not every time but by far the majority.... ICP lower P than Hanna Hi-736 ...Retest the retain...looks good.
Call me a slow learner ....It took me almost 2 years to stop and ask the question what is going on? I am either the worlds worst (but consistent) Tester or there is something I am missing. I had seen on this forum several discussions about the same observation...ICP tests lower then Hanna Tester and didn't really follow them until
@taricha made a statement that caught my attention "maybe the sample is not stable" ...That set me down this "Rabbit Hole"....And literally 100's of tests later here we are...My first posting (
https://www.reef2reef.com/threads/sample-storage-and-its-impact-on-phosphate-measurement.696800/) was the beginning of my trip to "Wonderland" but only the beginning. I basically set our to understated what impact storing a sample has on my test results... My basic conclusion for myself was not to store samples for any length of time in any type of container....Which is as it turns out was well known before I went down the "Rabbit Hole"....My bad should have done more looking...none the less I got really good at the test, ....
But it still did not answer my question "Why do my ICP tests come out consistently lower than my Hanna results?" According to my understanding this should not be happening and if I read the paper you provided correctly they agree...Should not be lower only higher or at best equal to.
I proposed in my first post, where I reported the depletion of P (PO4) during storage when tested with a Hanna Checker, that this depletion could be an explanation of why my (as well as others) ICP tests for P were lower than my Hanna Checker....This speculation has taken me down an even deeper "Rabbit Hole"...on containing Biofilms...Metal immobilization by biofilms and all kind of exciting stuff. My work on this second "Rabbit Hole" is almost completed and it was focused on the "Missing P" issue and how I might correct it....
I do not characterize this a an ICP testing issue, but an overall protocol issue relating to the sampling . As the information you provide and the article I have suggest water sampling is no trivial issue....Can't just "dip and go"....One has to consider type of container, preservation requirements etc. based on the elements you are testing . As you very well pointed out some elements and compounds are "rugged" and can be handled accordingly but others like P...PO4.. NO2 are much more demanding in the sample handling...This is like a whole science unto itself!
Does ICP testing overcome these sampling issues? Can a vendor hold a sample for several days and still get the same test results?...I can for sure say my testing methods can not. I contend that this sampling problem does impact ICP test results for some elements. Some how, way beyond my understanding, it has to do with Bio-activity and biofilms that prevent the availability of some amount of P in this case from being available to the current test protocols...This is the focus of my second trip to "Wonderland"
So this is a very long winded way of saying ....I am not quite yet settled on the matter.
Respectfully
rick
At least I had that in my old Lab 
