When they say the spell "is a bard spell" (fro exemple) they really just means that it interacts with class features that call out 'bard spells' (or Cleric Spells or what have you). It also usually means it uses the same rules at your other spells in regard to casting stats and preparation rules.
How do they want it to interact with spell versatility would be my question. Either those spells work with it just like any other bard spells, or they are restricted and spell versatility doesn't work with all the bard spells known even though they aren't any different from other bard spells known.
I can see the argument to push "the" instead of "your" bard spell list but I would have a hard time enforcing that level of technicality on it.