That advice doesn’t make any sense to me. I subscribe to both Adobe Stock and Shutterstock, and I can attest that neither service provides a proof of purchase. There is nothing to save. No transaction number, no invoice number, no nothing.
Both services however, do compile and provide detailed account information to customers. On each of the sites, I can see which photos I purchased going back years.
I have no reason to believe that the stock services would not stand up for me if someone should incorrectly accuse me of pirating a stock photo. The stock services' business model would go down the tubes if they didn’t stand by paying customers like me.
I could add one category of photo that you are leaving out, by the way, and those are photos published and made available with Creative Commons licenses. I make extensive use of those kinds of photos, and the licenses and terms are very easy to read and understand.
They’re also very easy to find using Google photo search, which can filter by Creative Commons license.
It’s necessary to verify that the photo is actually licensed under Creative Commons, of course, but if it is and you follow the terms of the license, you’re golden.