The ones that are either white or green, and if you mouse over them, I've seen "weak," "standard," and just now "superior." What do these mean? I've been trying to figure it out, or see if this question has already been answered, but with no way to search through topics that's rather difficult, and I can't seem to figure out any kind of major pattern or correlation with them.

the colors of the star indicate the number of trait the pet have

White = no trait ( obviously i will not color this one)
green = 1 tp
blue = 2 tp
purple = 4 tp
yellow = 6 tp
red = 8 tp

1 tp allow you to get a carry trait and 2 tp a visual but some pet only have carry so it's not because the stars are blue that they have a visual, they can get two carry instead (it work with every colors who have more then 1 tp)

ps: you nead a trap of the same colors or stronger to catch a pet

and the number of star is the difficulty of the encounter.

* = week
** =stendard
*** = superior
**** = elite
*****= boss


hope that help you ^-^

Thank you very much!

Just going to add, the number of stars also tells you how much base mana a pet will have if you catch it.
Excluding boss level (which cannot be caught):
* = 60 (example)
** = 80 (example)
*** = 100 (example)
**** = 120 (I haven't caught any, but you probably get the idea?)
(Fabled pets, which have naturally higher base stats, are ** = 100, *** = 120, **** = 120, 120 being the maximum possible mana)