Is a Masterful Snare trap the only trap capable of capturing Wild Sylesti? Or will a strong trap potentially do it just not as likely?

The Masterful Snare Trap is the only one that 100% guarantees that you'll catch the sylesti. The other ones, When you buy more expensive ones they can catch more rarities of pets. So the lesser one can catch Common and Uncommon then the Standard one lets you catch Rare ones.