I cant see the harm in printing out the actually email from the job that you have just applied for. I have been doing this for the past year now. Its my way of providing proof that I am applying for jobs, and it keeps them off my back. I also save every email for my record.
I have learnt over the years, it is best to cover your ass.
I did this for a while, but it was costing too much in printer cartridges. I politely asked, as the WP company had demanded I do this, if they would cover the cost of replacement cartridges, they refused and I certainly wasn't paying £45 for a set of cartridges just to print out such things. So instead I just screen capped each application, attached them all to an email and emailed them at the end of every week. I suppose it was a bit naughty of me to save the sceencaps as bitmaps instead of jpgs though, leading to my adviser moaning about the size of the emails because his IT department were having a go at him about it.
He now wants me to take snapshots on my phone and let him look through my phone pictures at my next meeting.
One thing the major jobsearch sites could do with is the ability to email you a list of jobs applied for at the end of each week, which you could then print out or forward to your advisor. That would be better for those of us that don't use UJM and don't want to use it.