Note that all of the nuances you mentioned are actually fairly light, and in most cases, come more from context than from the particular choice of phrase, so I don't think you really need to worry too much about somebody getting the wrong idea just from picking the wrong phrase here. All of these could be used in most situations without having a lot of extra meaning. That having been said:
Have you run today?
This has a bit of an implication that you expect that they have run or that they will run today (because otherwise, why would you be asking that question?). It's perfectly reasonable to say this if you already know that, for example, running is something they usually do every day anyway, and in that case it doesn't necessarily have much additional meaning (just "I know you usually run every day, did you do it today too?")
I think the reason that "Have you run" can sometimes be seen as encouraging or prodding is because it can also potentially be interpreted as "have you run yet?", meaning "did you do it already?". Depending on the context, however, this could either mean "Did you do it already? If not, you should!", or it could just mean "Did you do it already, or are you going to later instead?" (which is not particularly encouraging/prodding), so this all comes down to context and interpretation.
Have you been running?
This is actually asking if they have (recently) been running for some reason. "Have you run?" usually has the sense of somebody running for pleasure or for exercise, whereas "running" could just be because they were late, or needed to get out of the rain, etc. As mentioned, without "today" or something added to it, this phrasing also implies you're talking about recently (i.e. a few minutes ago, not hours ago).
The reason you say that it's said when you see sweating, etc, is, again, more one of context. If you didn't see those things, then why would you think they had been running, so why would you even ask the question in the first place? It's not the sort of question you'd just ask somebody randomly without some reason to think that they might have been running recently. However, it doesn't have to be visible results that prompt a question like this. You might ask this, for example, if you happen to know that they usually go running at a particular time (for example, they like to run during their lunch break, and they just came back from lunch).
Probably the most neutral form, which you didn't mention, would be the following:
Did you run today?
Again, this implies that you have some reason to believe that they would have run today (e.g. you already know that it's a habit of theirs), because otherwise why would you ask the question, but it doesn't have as strong of an association with the "yet" sense of things. It's really just asking whether somebody did or didn't run today.