I always thought
L. debelius was sometimes referred to as "fire shrimp" or "blood shrimp" and the common names "skunk cleaners" and "scarlet cleaners" both referred to
L. grabhami. Oh well, that's the problem with common names, they're all over the place.
Anyways. This is what I can tell you. Cleaners (um ... I only really think of "skunk cleaners" as cleaners) are really the ones that are the most bold of all three. I find peppermints tend to be shy and nocturnal, so you can rarely see them; and fire shrimp are either nocturnal or from deeper water (or perhaps both) and so are even more shy (i.e., you'll hardly ever see them).
Right now I have 4 cleaners in my 75g, they make a great group. The are not shy at all, they're great to look at, and they even form "cleaning stations."
Lysmata sp. (all three) are "simultaneous hermaphrodites" (how's that for a mouthful) which means that as long as you have two or more, they will breed. I've never tried fire shrimp but I would assume they're the same as peppermints and cleaners. Peppermints have for sure been raised in captivity, I've heard mixed stories about cleaners as to whether they've really been raised in captivity or not. Suffice it to say, unless you're prepared to specifically try to raise the larvae, the tankmates (fish, corals, the parent shrimp

, ...) will all have a spectacular feast the night the eggs hatch. (The shrimp carry the eggs until they hatch, which seems to happen shortly after lights-out.) It sounds kind of brutal, but on the other hand ... you couldn't ask for a better food...
Ummm let's see what else can I think of. They're all theivous little pests (I use the term affectionately though

) that will need to be well fed otherwise they will form maurauding (sp.?) hordes that steal food from anyone they can steal from. (Particularly from slow feeders, i.e., anemones, LPS). Have heard mixed stories about reef suitability of peppermints, but I've never had a problem with them myself.
Anyways these comments are based only on my experiences. My peppermints never touched aiptasia which was the original reason I bought them, so your mileage may vary if you buy them for that reason. I think, now don't quote me on this, but I think that there are two species of
Lysmata shrimp that are called "peppermints" and they look identical, except that they come from different places (maybe one is Pacific and one is Carribean or something like that), so there mmmmmight be cases of possible mistaken identity when it comes to peppermints. (Disclaimer: {As usual} I really have no idea what I'm talking about at the best of times, so don't listen to me...

)
Hope this helps though! cheers 8)