I have the PAL version and so I need to beat the game 3/4 times to get there. Question: do the time attacks (normal/hard) count as replaying the game?

No. But if you acquire the Lizard Stone from time attack, you can find the lizards with blue tails and increase your stamina that way.

Are there more than one set of lizards? The white tailed (or so it seems to me) lizards can be found at each save game shrine from the very start. The stamina boost is pretty tiny though.

Yes. There are regular lizards and white tailed lizards. The regular ones give you health, or something.
The while tailed ones repeat a few times. That is, if you kill one, you can return two or three times and it will be back.
I found killing colossi the best way to get stamina though.

white tails add grip normal do nothing fruit add health