No, no, no, no. Perhaps this is my fault. It is difficult to understand, so I'll try different methods until it clicks.

James - If you say, "Let's play in A minor", you might as well say, "Let's play in A aeolian", because it's the same thing. When people say they're in D major, they could also say D ionian. It's the same thing. Ionian is the modal term for 'major', and aeolian is the modal term for 'minor'. While all the modes can be classified as major or minor (or diminished in poor old locrian's case), it's because of their interval qualities.

If something is in A minor, that is the key it's in. The mode it's in is A aeolian, because when talking about modes, we want to use modal terms. Don't say it's in C aeolian - that's a whole other key. A minor comes from C major, because A aeolian is based off the sixth note of C major (or C ionian - same thing), and likewise C major is based off of the third note of A minor (or A aeolian).

I think what's confusing people is the use of the terms 'minor', amd 'major'.
When talking about what key a song is in, we use the terms major or minor. For instance, Happy Birthday is in E major (or whatever key you sing it in - the point is that its straight major). Modes really only apply to guitar. No other instrument or form of music than guitar music uses the concept of modes. In classical music of the past, songs were either major or minor (ionian or aeolian, although they didn't use modal terminology). In rock or metal, etc., it's not always the case. 'The Shortest Straw' by Metallica is in E Locrian. I suppose you could say it is in the key of E Locrian (don't say it's in F major (the relative major of E locrian) - it's just plain confusing - the root of the song is E).

Music was fine until guitars came along and threw everything out of whack.