"Es ist eine Vorstellung die von einem Freund arrangiert wurde" definitely sounds like "It's a show that was arranged by a friend", and not about an introduction (in German). For an introduction between two people you'd rather say "Es ist ein Treffen, das von einem Freund arrangiert wurde."
Although on this page the translation is given as "This is my last show" when I wrongly translated "Vorstellung" as "impression", Duolingo gave as the correct answer "That is my last imagination." This seems even more odd to me than "my last introduction." I wondered if the context for such a translation would be at the last second preceding one's death. I have noticed that Duolingo frequently gives an incorrect response after I give an incorrect response. This doesn't help me learn from my mistakes, unfortunately.
an "Idee" (=idea) is just what you have in your head. There are other words in German, like "Einfall" (~a sudden thought) or "Vorstellung" (= imagination), that cover parts of what you call an "Idee" which you sometimes can use synonymous. But "Vorstellung" has other meanings as well, like "presentation" or "show", that have nothing to do with an idea. It's just the same word for a different concept, so you have to look at the context which meanings makes sense.
So "Vorstellung" gives me the impression of "putting something forward". I can see that in terms of introductions, where you might literally put someone out in front of you to introduce them. And I can see it as a show, where it's a production you present to viewers.
I'm not sure how that plays into "imagination" and "picture", though.