How do I choose a contemplative lay order? And do I have to be Catholic to do so?

A reader recently wrote to me:

Dear Carl, I grew up as a Protestant, but as I’ve journeyed into Contemplative Christianity over the last while I’ve become more amenable to Roman Catholicism. I have been wondering about lay orders and how to connect with the contemplative tradition through a lay order. I was wondering if you could shed some light on which lay orders there are (main ones) and what the main focus is in each one, and specifically which one, if any, “carries the most weight” of the c...