After having spent years and years of putting my hands in stuff that I shouldn't have without any repercussions it has finally backfired on me.... now, I get reactions without knowing what caused them even! Very frustrating for sure, never mind painful and or annoying.
Regarding to your reaction not differing between gloves vs no gloves... Your "no gloves" reaction has too many variables... what products and what applicators did you use?!?!
The second reaction you had while wearing common rubber gloves on the other hand raises a flag for me as my wife has the same problem.... namely latex, commonly used in kitchen rubber gloves. She also reacts to wearing rubber boots, band-aids with latex based adhesive and so on. In your case, perhaps you were using a foam type applicator when not wearing gloves that may have contained latex or perhaps something you were cleaning, such as floor mats or what have you, might have contained latex and thus caused the same reaction.
Of course, I'm not a doctor, even if I did spend the night in that hotel chain advertised on TV, so I'm not diagnosing anything but rather just tossing out some ideas that came to mind.
As someone mentioned, try a pair of the nitrile gloves (mostly blue colored if I recall correctly) or even a pair of vinyl gloves, just to eliminate the latex issue. You could also try some tight knit cotton gloves if you're not getting your hands wet as some people (my wife again) react to just having their hands inside a non-breathing glove! Different materials have different resistances to chemicals and so I would suggest you do a google search for "Glove Compatibility Charts". The page that came up at the top of the list has a long list of links to such charts at the bottom of that page and that would make an excellent starting point for you.
For what it's worth, I've not had any particular problems with detailing chemicals but I have stayed with a couple of company's for the most part when the first couple of items didn't cause me any grief and so far so good (Meguiars/Collinite in my case).
Try to test only one or two things at the most and let a day or two go by if all is well, and then try one or two more until you find out what exactly bugs you.....
Regards
Christian