I’m probably not going to do anything about it, but my 92 XJS V12 with the “light up” version of the wiper stalk will spray fluid on the windshield and wipe a couple of times about 50% of the times I start the car. Yes, I’m almost certainly brushing the stalk slightly when turning the key and that causes the couple of wipes… but the sprayer is a push in button on the end and it’s impossible I’m hitting that.
Anyway I just thought it was funny and I couldn’t find a similar post on the archive.
Funny. Don’t ya just love the quirks of XJS’s. I normally don’t take mine out in the rain since I don’t use it any longer as daily driver and no longer drive to work, so wipers not used much. Friday I decided to drive in the rain and was relieved to see the wiper wash function spraying without being clogged 
Gotta ask…what is “light up” version of the stalk mean? Mine is LHD and the wiper switch is on the right side of the steering wheel just above ignition. So I have done the same. More often though, trying to turn on the right hand turn signal on my Chevy Silverado I accidentally switch on the wipers. I hate that when the windscreen is dry!
I’m betting on a ground issue.