Instead of using the language of spells known or prepared, just directly say what you want it to allow: casting Revivify, nothing more, nothing less. It doesn't hook into your class's spellcasting feature in any way other than a way to expend a spell slot.
Ring of Revivification
Ring, rare (requires attunement by a spellcaster)
While you are attuned to this ring, you can cast Revivify using your own spell slot of 3rd level or higher. The material component must still be provided.
Your spellcasting ability for this spell is the same as for your spellcasting class.
I debated whether to include "as an action, you can cast..." I chose to leave it out to avoid any implication the action might be Use Magic Item rather than Cast a Spell, which would have implications for casting in the same turn as a bonus action spell.
Narratively, you might think of it as the caster channeling their magical power into the item to produce the effect (rather than the item granting knowledge / ability). But mechanically we want it to still work like casting a 1-action spell. I think this wording implies that.
This avoids letting wizards copy it into their spellbook, because we avoid using the language of "prepared spell". The language is similar to feats like Fey Touched (but without the first cast being free), or Eldritch Invocations like Sculptor of Flesh
(You can cast polymorph once using a warlock spell slot. You can't do so again until you finish a long rest.)
I considered writing "you can cast Revivify using a spell slot" or "you can cast Revivify by expending a spell slot of 3rd level or higher". The latter has more room to be misread as being a different action than Cast a Spell. (I say misread because I think "you can cast" still implies that you have to cast the spell yourself.)
The clause about spellcasting ability is I think only relevant if using Matt Mercer's "Fading Spirits" optional resurrection rules as used on Critical Role, published in the Tal'Dorei Reborn book. Revivify involves a roll which is like an ability check, but strangely phrased to not quite be one (perhaps so guidance and inspiration don't work, nor does Jack of All Trades or a Stone of Good Luck)
The caster makes a quick resurrection check by rolling a d20 and adding their spellcasting ability modifier, against a DC equal to 10 + 1 for every time the character has returned to life before.
Molot's answer makes a good point about other features you might want the ring to have. That's compatible with this answer. Another option would be to make it non-attunement, although that opens up the possibility of micro-managing it when someone's low on lvl3 slots, or taking it off the dead guy's finger and using it to revivify them.
A non-attunement ring still has the tax of someone keeping a lvl3 slot unused, which is significant for a low-level party. A high-level party with lots of other good magic items would probably prefer non-attunement.