There's nothing wrong with saying that you have a better offer. In fact, I feel that it's a suitable reply, since everyone wants to have career advancement, and if another company can offer you that, then that's the truth you should tell.

An industry is usually small, and people talk. Even if you don't divulge on why or where you are going, things have a way of getting around.

I've been through similar experience where I openly say that I'm moving to another company (unless if it's a competitor where you may have to be a bit more discreet) and I've been rehired at one of my old company, and a job offer from another one of my old company.

A formal letter is always the best. I believe you need to either hand in or scan the letter to HR upon notifying your superior anyway, so a letter would work best in this situation.