Are we talking a real refugium, with sand or mud, rock, assorted creatures, or are we talking an area for algae growth? Two different things where the term "refugium" is used interchangeably.
IMHO they are both refugiums, just slightly different end goals in mind. It's kind of like 'Live Rock'. Are we talking rock with good bacteria for processing ammonia and nitrite or rock covered in all kinds of 'other stuff' like coraline algae, other algae, sponges, feather dusters and more. They are both live rock, just slightly different end desires. Sumps and refugiums can be very useful, but they are by no means required.
I had a 65g shallow reef (4'x2'x1') and a 25g frag tank that were both tied into a 125g sump/refugium (80% refugium, 20% filter socks and skimmer). The fuge was 30% chaeto and 70 live rock as there was very little rock in the tanks. I never had issues with nitrate or phosphate. Almost everything I tried in the tank did well... almost. And because I live in SW Florida I would bring home all kinds of critters from beach walks along the Gulf of Mexico or from snorkel trips to the Florida Keys.
But at the same time, I currently have downsized from 4 tanks to just one. And for the past 2 months the only thing my 50g cube is attached to is a Brute trash can with a filter sock, 25g of water and a return pump, not even any live rock and no skimmer! However that will change as I've made a new sump/refugium (40g) and ordered a new skimmer. But as it is, the stuff in the cube are doing just fine. I have lots of zoas, a few lps and sps corals, 30+ rock flower anemones, some RBTA and maxi-mini anemones, a few feather dusters, shrimp, fish, serpent stars, 2 different sea cucumbers, snails, crabs, 2 small urchins, macro algae and 2 mangroves!