What is the official/correct term for step down triggers on a controller

I am trying to buy a gaming controller and I have always referred to step down buttons as the L2/R2 triggers(the lower triggers) that are positioned down and back then the standard controller would have them. These have always been more comfortable for me as controllers and I have always refereed to them this way but I can't seem to find one online now that I am trying to buy them. I'm looking for something not to pricey but is a decent controller like $20-60 dollars that has these step down triggers. But apparently with out the right terminology I cant seem to find them online anywhere. I wanna say the last one I ever used was a Pelican controller on the PS2 but if I buy one of those is it PC compatible?

-Edit-

i.ebayimg.com/images/g/Al0AAOSw8gVX8lDw/s-l1600.jpg

This is a picture of the old PS2 controller with the buttons im talking about. Controller type is the old PS2 Pelican chameleon controller. But i need something i can Use for PC that still has buttons like this.