Should I rinse with water if we're using the same glass for a tasting of several wines?
It's not necessary, by any means. In the context of professional tastings, water-rinsing is never done. At least I don't come across it. Simply pour out (or sip) as much of the previous wine as possible, then follow directly with the next. Chlorinated tap water is no better than residue from another wine, in any case.
There is an exception: When moving from red to white, which will leave a trace of colour in the latter, undesirable mainly from an aesthetic standpoint. But a tasting normally progresses from white to red. If you're afraid the flavours will cross over and you want to be a stickler, you could "season" your glass with a small pour of the wine to be tasted, then pour it out, adding a fresh pour of the same wine.