Agreed. Crisis on Infinite Earths is pretty much a prerequisite to Infinite Crisis. At the least one should read a sturdy recap of CoIE before reading Infinite Crisis because it's totally a sequel.
Final Crisis is not part of either of those stories. It's one of many universe-wide events and at DC those events are commonly tagged with the "Crisis" label. It's a callback to the first JLA/JSA team-ups when some "Crisis" would cause them to team up though they existed on parallel earths.
There are also lots of events in DC history that were "Crisis"-level, line-wide events, like Zero Hour (too old and dated to really enjoy now) or Blackest Night (highly recommended), but didn't include the word "Crisis."
In fact, publisher Dan Didio wanted to name the very recent line-wide crossover event, METAL, as a "Crisis."
And then there's Identity Crisis which deeply touched all corners of the DCU (highly controversial - people love or hate it - I love it) but which was a much more intimate story than the other "Crises" in terms of the way it was told. The others are all pretty cosmic. Identity Crisis was an emotional story that included personal trauma and crises but it doesn't at all belong with the mega-events like CoIE or Infinite Crisis or Zero Hour or Final Night.
For purpose of this post I went searching for a comprehensive list of DC's major event stories and this is a pretty thorough one. I'd give it a look if I were you, OP. Hope it's a help.
http://dc.wikia.com/wiki/Major_Event...he_DC_Universe