I think the two kinds of logs were introduced explicitly for quest purposes, but appeared in the game a bit earlier than quests did. As far as I know there is no difference between the kinds of logs apart from quest suitability.
In real life cutting a tree while it's raining or dry makes no real difference. It's more important to protect the logs from getting wet after they're cut down than a little rain during the cutting. One important cause of damp in wood is from the ground, in particular if the log is lying in deep moss, so getting a space between the ground and the log (i.e. by placing it on thick branch pieces) is probably as important as providing a "roof" above it. You should also remove the bark from the log both to get it to dry faster and to keep tree burrowing insects in check.
It can also be mentioned that one historical way of transporting timber for cutting into boards (much later than the UrW timeline) was to dump them into rivers and tend the logs (pushing them out again as they got stuck, dismantling pileups, etc.) as they floated down river towards the lumber yards, and the timber didn't take much damage from that treatment.