The mechanic of having a player's health be represented through some other abstraction than a bar or counter is common, but doesn't really have a specific standard term. We just label it as what it is, in English: a health system without an obvious bar. There's lots of different ways you could slice it, depending on the specific aspect of it you're trying to convey as important.
For example, if the important thing is that there's no easy-to-read discrete indicator of your health as a percentage or value, you may say it's an "opaque" or "obscured" health by system.
If the important thing is the visual effect you replace the discrete readout with, you might come up with a visceral name like "blood barrel" (which, I have to say, is not a term I've ever heard before for the concept in the industry).
If you think the important thing is to be going for "realism" (questionable) by making it hard to tell precisely how close one is to death, you might call it a "realistic" health system.
Generally the importance is probably with the "no UI" aspect, so you'll most often find variations on that theme. "Distraction-free," "in-world," "UI-less," "immersive" (also questionable, as your eyes are not a camera lens as they are in that screenshot). And so on.
To address some comments:
Isn't the most important aspect of this system the fact that there are
no medkits or health packs to pick up?
Based on the original text of the question, no, because a system where health can't be restored by consumables is orthogonal from one where the feedback about your current health state is communicated without discrete bars or graphs, as in the screenshot and descriptions posted by the OP.
this type of health system is well (perhaps even universally) known
as "regenerating health".
Yes, a system where your health regenerates is fairly universally known as a "regenerating health" system, but that doesn't appear to be what the asker is talking about. The description and screenshot illustrates the mechanic where blood smears (or similar fuzzy metrics) convey how close the player is to death, and don't speak of regeneration specifically (just healing, which is more general).