In the UK [or at least my part of it], a tea towel is a drying-up cloth made from cotton or linen or a mix of the two ["linen union"]. Linen leaves less lint behind on the crockery/pots & pans, and is particularly good for glassware; they are often called glass cloths on the red/green/blue & white ones we have - stripes of colour along the long sides, and a band down the middle with the maker's name and "glass cloth" or "linen union" or similar writing in the weave.
Even though dishwashers are much more common now, I still prefer to wash glasses by hand (especially the nice/lead crystal ones), as I've never seen any that go through a dishwasher regularly that don't get that 'bloom' of scratches. It's probably our water.
Again, where I'm from "kitchen towel" is the paper on a roll that looks like a giant toilet paper roll. Just "towel", implies a looped-weave towelling fabric hand drying cloth. Ironically, of course, we call the towel in the kitchen a kitchen towel, too!
I'm guessing way back in the day, tea towels may have had something to do with the meal "tea", but doubt they are linked to the beverage. Over here, they are a popular gift for people you don't know what to buy for as a present from a holiday, or housewarming gift. Not everyone likes them, because I quite often see unused ones in charity shops [if they are linen, I snap them up; linen doesn't last as long as the cotton ones].