I would never feel embarrassed having an obscure rule (like some on this forum) pointed out to me, nor is it rude to do so. Contacting them privately often is not read in a timely manner, and thus the point is lost.
Publicly quoting the rules is not backseat moderating, period.
You may not see it that way, but I've found that just about everyone else does. I don't see why time is an issue though? Isn't the point to make sure they know not to break the rule again? In that case, why does it matter how long it takes for them to learn this?
Honestly, reporting is better anyway. If time's an issue to you, then reporting is the best solution. A moderator will see the report thread (assuming that's how they've set them up) or email and hopefully they'll let the member know what they did wrong.
You may not see it as backseat moderating, but a lot of people do. I'll agree that some people actually quote the rules to help people, but most don't. The person's intentions don't really matter anyway, though. Like I said, it's often perceived as being rude. Why does a member need to remind another member about the rules? It's none of our business; we report and then go on. We do the same thing in game, so why not do it on the forum too? I'd compare backseat moderating to having a sibling that you're close to in age try to discipline you. No one ever likes it.