I do not see the second sense being the same as the first.
First sense:
Although the sun was shining it wasn't that warm.
I take this to be the equivalent to:
The sun was shining. Yet, contrary to what you might assume from that, it wasn't that warm.
In other words, although is being used to point out a conclusion that is contrary to the evidence.
Second sense:
He says he has the team shirt, although I've never seen him wear it.
I take this to the the equivalent to:
He says he has the team shirt. Oh, and I've never seen him wear it.
Here, although is not being used to state a conclusion that is seemingly in contradiction to evidence. It's being used to state something additional and related, but nonessential.
Which, by the way, is the reason for the comma: what follows it is a nonrestrictive relative clause. The essential meaning of the sentence would not be changed if the comma and everything after it were removed (the resulting sentence would still be grammatical).
In the first example, the sun is warm; therefore, it's natural to assume that it is warm because the sun is out. But that's not actually true. So, although is being used to point out an exception.
In the second example, no conclusion is being drawn from the fact that somebody has a team shirt. Just because somebody owns a piece of clothing (or is storing it somewhere) that doesn't mean that they should necessarily wear it. I own several pieces of clothing that I haven't worn in years. Although is not being used to point out an exception at all; it's being used to simply mention a related, but optional, second fact.
To specifically demonstrate the syntactical use of the comma in all of the example sentences of the second sense, here they are without the nonessential information:
He says he has the team shirt.
I don't believe we've met before.
I think his name is John.
I'd love to have dinner with you.
The book had a good plot.
Note that in the last example sentence, the nonessential information is parenthetical, so the last word remains.
Last, note the syntax in the first sense where although is used as the first word in the sentence. In the second sense, it always comes in the middle of the sentence. It's this syntactical structure that determines its meaning.
First sense (contrary to an assumption):
Although the sun was shining it wasn't that warm.
Second sense (nonrestrictive relative clause):
It wasn't that warm, although the sun was shining.
Best Answer
Always taking a bit of a chance using the Internet to answer an Internet question, but Daily Writing Tips says that not only are while and whilst interchangeable, but that in fact while is the original version. The very authoritative Michael Quinion backs this up on World Wide Words (and that is a site well worth visiting for anyone interested in the English language).
So I think your question proceeds from a false premise: they haven't come to be used to mean the same thing, they do mean the same thing, and there is nothing wrong with using while in all cases. (And to answer the question that wasn't asked, since it is shorter than whilst and clearly understood by US as well as British English-speakers, I'd say stick with while every time.)