This is a specific case that overrides the general one
Firstly, on PHB, p. 7:
This book contains rules, especially in parts 2 and 3, that govern how the game plays. That said, many racial traits, class features, spells, magic items, monster abilities, and other game elements break the general rules in some way, creating an exception to how the rest of the game works. Remember this: If a specific rule contradicts a general rule, the specific rule wins.
Generally speaking, magic staves (and other magic items) that allow you to cast spells, usually "once per long rest" or "once per dawn", or through the use of expending charges, do not require that you have those spells in your spell list.
The Staff of Defense and Spider Staff, however, both override that by stating "if the spell is on your class's spell list" (LMoP, p. 53), as you note. Therefore, these staves do require that you have access to these spells already, which is different to other magic items that allow the casting of spells.
As @V2Blast points out, the Starter Set came out before the PHB or DMG were released, so such design conventions had likely not been finalized/refined at the time of its release. It may be that these are different from the items in the DMG because they decided to change how such restrictions were implemented, using attunement requirements rather than requiring you to already have access to the spells that a magic item provides.