Some commentary focuses on quarries and industrial lakes. I am not of the opinion that that makes a difference: what's down there (ie, machinery, rock ridge formations, etc), and what's in the water (ie, chemicals) is more important than how deep the water is. In this case, signs should scream "Danger! Keep Out!", rather than "Danger! Deep Water!"
I'm also not of the opinion that cold water temps at depths are a reason for the signs, although I do think it might play a role in swimmability, which pertains to your initial question. But experienced swimmers know this, non-swimmers do not. And at depths where temps make a difference to non-swimmers, the depth - and not the temps - is a more troublesome issue for the non-swimmer.
Although the question is broad, you did give an example of a lake. In this case, things like currents or animals/fish playing a role in the "deep" part of the "danger" is less likely.
The word "deep" doesn't even have legal definition. After all, pools are often labelled "2 feet deep", "12 feet deep", etc. Is 2 feet really "deep"? I think not. So a sign that says "deep water" is simply playing to fears, forcing people to be careful.
And that is a valid context: to play on fears, as well as to avoid legal liability. For example, you mention a lake. Lakes are not "x meters deep"; rather, they have variable depths. Someone who cannot swim, perhaps a child, may walk into a shallow area of a lake, walk around, and suddenly find themselves walking at depths they can no longer walk without having to swim. Is it deep? Yes, for that person, it's too deep for them to walk. Is it deep for a diver? No, of course not - but a diver is used to temperature changes as well as knows how to swim. The signs, therefore, must pertain to those who cannot swim.
Further, you see signs at public pools which indicate how deep that part of the pool is - but you never see signs that warn of "deep water", even at the 4 meter end. There's a pool near me which has a 20' depth for scuba instruction and practice. No signs. Why? Why does a lake with variable depths get signs, while a pool with a known depth not get a sign?
And that's the answer: the depths vary, and will be too deep for non-swimmers. The water isn't more or less swimmable at varying depths, it is that it is less walkable at varying depths.
And beaches: why do beaches never have signs indicating that water is deep? The depth most definitely varies. There are currents. Temperatures more assuredly change. There are fish, some really nasty ones like sharks and jellyfish. Yet, there are not universally placed signs warning people of these dangers, right? Well, not true: It would take away from the beauty of the beaches to have these signs all over the place; but in fact, they do place signs there. And they do tell you of the dangers: deep water, rip tides, shark sightings, rocks, buoys, and so on. They're just usually not placed right in the middle of the water where one might swim (or walk) into one. And they're very specific about the danger that lurks.
As to liability: That there are signs at all is probably to avoid legal liability, but only in limited cases. The reality is that lakes are typically owned by government, and suing a government entity is a herculean task. I think, then, liability is less of the impetus to place a warning sign then it is to help people know that waters are probably deeper than they are tall. In private lakes, perhaps a warning sign has more legal implication, but your question didn't distinguish between private and public bodies of water. Rather, you asked if depth made water more or less swimmable.
By the way, the example link you provided struck me as odd (and comical, with very young children playing around a scary looking danger sign). I'm from the US, and am wondering if my definition of bathing and paddling are different than in the UK. I'm imagining that bathing has a connotation that there is a lot less formality with regards to swimming, and deep water can affect your bathing. A lost bathing suit can be most embarrassing if you can't swim down to retrieve it - but that hardly warrants a "danger" sign. As to paddling, well, you're in a boat; the article mentions that swimming and navigation are two separate distinctions in UK law, and that might mean the difference. Paddling, also, is dangerous to swimmers who could become injured by the craft in which you're paddling, or the paddles themselves.