So I'm about to get into using lesser pet dye, but I need to know something- why does it say that it will only be fairly accurate? Does this mean that the color will change to a slightly lighter or darker shade? Or something else? I would appreciate if someone could let me know!

Yes, the color you choose will not be the exact color that your pet gets dyed. If you choose pure white then it may be gray, light pink, off-white, etc. It's a matter of luck, really.

Yep, lesser dyes can be quite inaccurate especially for really light or dark colors. You could be going for a dark green and end up with dark purple instead. It is the same when generating a pet using a regular essences. I have had pretty bad luck with lesser pet dyes so I almost never use them.

I have had the best luck with blues, but I think that is because I am not too picky with blue. :-P