No I don't, but yeah this is correct.
Bitcoin: The title.
bitcoin: The unit.
BTC: ISO code.
₿: The symbol.
That's the least grammatically issues we have on this forum, though...

Yes, because here in the forum, doesn't matter how you use the word bitcoin in your post. And everyone understands what you are referring to. But maybe, if you are writing a professional article to be published, you need to be aware what you are writing as most people that may read the article don't have the knowledge what you're talking about even for us is considered a popular term and also to observe the proper use of this word following grammar rules.