The intent is that a player should not hear “No effect” based on needing a specific carrier or type of weapon to administer the killing blow, so it should always “work” on the field. Whether it actually works long term because of a monster specific “may only be permanently killed by...” clause is up to the local plot committee.
Things like Vampires might have universal similarities, but they are also very local.
Personally, as a writer I will probably do something like “tier 1/2 are permanently slain by slayer/stake, tier 3/4 are permanently slain by stake, and tier 5 is special means. Tiers 3/4 killed by Slayer should be considered for recurring personal nemesis role, and Tier 5 hit with the negative Slayer RP means a player bought a ticket if the villian learns who did it.”