You say "Shouldn't Cost-Per-Impression be CPI?". You'd think so, but the 'M' in CPM stands for 1000, thus its referring to paying for 1,000 impressions of your ads, whether they get any clicks or not.
That's what you're buying, impressions. And there's more involved than just bid x clicks. If you're getting a good click-thru ratio (CTR) then maybe CPM is for you.
One thing you need to Strongly consider is, "Are you going to be watching your campaigns closely?" If not, then don't use CPM, unless you've got deep pockets, or have really done your homework.
Because you're paying for impressions, whether you get clicks or not. And if you picked a keyword that's more popular than you expected, those impressions costs may very well equal a house payment.
But if you can watch it regularly, and get a good CTR, then this just may be your gold mine.
The videos below help explain the differences, and how CPM is calculated, along with a special offer in Video 2.