Exactly!
Longer more drawn out way of saying "Yes:"
You're right in how you understand it. The most confusing part is that up until next year, the AHL had less teams than the NHL which lead to some teams doubling up their affiliations, or NHL teams spreading their prospects among teams without affiliating.
If the, for example, Wild wanted to end their affiliation with Houston, they could take Robbie Earl and whoever else was under Minnesota Contract and place them wherever they affiliated next.
Houston only players (ie: Tony Hrkac) are still under contract with that club and could only be moved by Houston. Minnesota doesn't control those players.
There is ALOT of turnover at the AHL level, much moreso than at the collegiate level (well, most college teams don't see this much turnover).
Springfield, will be on their third NHL affiliate since 2004-05.
2004-2007 Tampa Bay Lightning
2007-2010 Edmonton Oilers
2010-2011? Columbus Blue Jackets
As for the affiliations, some AHL clubs are owned by the NHL team and place themselves wherever they da** well please. (Read: NJ Devils own the Lowell Devils, Minn. Wild own the Houston Aeros, Dallas Stars own the Texas (Austin) Stars). Other AHL clubs have affiliations with a local ownership group (Chicago Wolves own their club, affiliate with ATL; Grand Rapids owns their own club, affiliate with the Wings; Syracuse owns their own, affiliates with Anaheim/Columbus/whoever).
For teams that (up till this year) didn't have any affiliates, they would place their players around the league. For quite a few years, the Florida Panthers and Buffalo Sabres shared the Rochester Americans. Both teams would supply prospects with Rochester signing only a few extras needed to fill out the roster. Last season, while the Austin Stars team was nearing completion, Dallas placed their players with the Chicago Wolves, Manitoba Moose, Grand Rapids Griffins, Rochester Americans, and one or two other teams.
This season, Anaheim (after their affiliate was forced to quit due to breaking league rules) placed their players with the Manitoba Moose, Abbotsford Heat, Toronto Marlies, San Antonio Rampage, and the Chicago Wolves.
Next season, all 30 NHL clubs will have their own AHL affiliate with no sharing, or doubling up. First time that has ever happened in the AHL.
I won't get into the ECHL affiliations with NHL clubs, because that's even more confusing with how they share many NHL and AHL clubs.