No.
Those Shield Cantrips are not different spells. It's one spell available to several different traditions, which you slotted more than necessary. The wording of the Shield Cantrip says "...you can't cast it (the Shield spell) again for 10 minutes."
That'd also be no different than if you slotted Shield in every Cantrip slot you had for one tradition. Yes, they're different slots, but they still remain the same spell, Shield. Sorry.
It's perhaps easier to imagine if Fireball had a clause that said after casting Fireball you couldn't cast Fireball again for 10 minutes. It's pretty obvious it wouldn't matter what tradition those Fireballs came from, all Fireballs would be unavailable. There isn't a "Divine Fireball" & an "Arcane Fireball", etc.
As a tangent, the innate Shield Cantrip you've listed has to fall into one of the spell traditions too.
The "it" references "spell" from earlier in the sentence.
Which spell? The Shield Spell.
It being in many slots doesn't make it many spells. It's just the same spell memorized many more times.
It being in many traditions also doesn't make it many spells. It's still one spell, the Shield Cantrip.
"That specific Cantrip" doesn't work either, since that Cantrip specifically is still Shield (as opposed to Produce Flame), which is exactly what the other Shield Cantrips are and which the spell says you cannot cast after blocking. It'd have to use much more explicit language, i.e. that Cantrip spell slot, though other Shield spell slots can still be cast.
When you have to argue semantics this hard in PF2, you have to step back and realize you're likely looking for a loophole that's either not there or not meant to be there which RAW says should the be closed by the GM.