I have tried to capture Sylisti, but many times it doesn't work. I have noticed that my traps (which are lesser snare traps) are failing on Sylesti with TWO stars. Do lesser snare traps only work on one star Sylesti, or are two stare Sylesti just a rarer catch?

lesser lets u catch up to uncommon
standard up to rare
strong up to epic
masterfull catches anything

also bring its hp as low as possible, it might help but I'm not sure
I alwats bring their hp down anyway, kind of used to it from other games

Currently, the number of stars has nothing to do with what trap you use to catch a pet.
The colour of the stars is what indicates what traps can be used. Lesser traps can be used on white or green stars and so on.... Basicly the colour of the glow on the trap is the highest rarity of pet it can catch.
The number of stars is the pets difficulty in battle.

Also, traps have a low success rate, so it is normal for a trap not to work. It can take alot of attempts to catch any pet, from one to... however many.