The Pinocchio Paradox "Now" isn't a reference to time duration but reference to a lying utterance (i.e., my nose grows in response to this sentence). — Soylent
I am not sure exactly what you mean here. If you say "now" is the cause of the growth then it is obviously false because he has not said anything before this to cause it to happen. So his nose will grow for lying.
Would there be any difference if instead of using "claims" one used states or says?
Premise 1. Pinocchio's nose grows if and only if he states any falsehood
Premise 2. Pinocchio states "my nose grows now"
Premise 1. Pinocchio's nose grows if and only if he says any falsehood
Premise 2. Pinocchio says "my nose grows now"