You are reading it right, but you'd have to be in a bit longer than a single day to obtain any of the required documents they accept as proof.
Agreed, I think I was after six weeks of basic when we passed out of phase 1 that we were told we were finally, officially part of the armed forces. There are exceptions of course, one recruit that I knew of was invalided out due to breaking his back in week 2 and he was looked after (i.e. he's regarded as a veteran and got some ongoing support), but that's quite a special circumstance.
As you say, you can't simply sign up to the reserves, resign a day later, and have any documented proof of service. We were told that if we bailed out of basic training
voluntarily before passing out at week 6, we were not considered as having completed the joining process.
You were in a trial phase if you like, if HM forces didn't like you, or you weren't keen either, it was a handshake, no hard feelings, and good luck in your civilian life. There was no certificate of service.