James: The spell remains a 2nd level spell, but you must prepare it in a 3rd level spell slot. Only Heighten Spell actually increases the spell level. I think that is what you meant, but it reads otherwise.
Sineater: Yes, the spell you extend must be prepared in a higher level slot, or in the case of a sorcerer/bard it must use up a higher level slot. The spell's actual level does not increase, however. Heighten Spell increases a spell's level.
Except for sorcerers and bards, the extended spell must be prepared at the normal time in the higher slot. Sorcerers use metamagic at the time of casting, and must have the proper higher level slot available, but it takes more time to cast (1 action spells take a full round to cast like a full attack action).