I really like what Donald said about the log
I think it totally wants to be based on whether or not they are using the VP counter. With VP counter, show the whole log. Without it, show each player's most recent completed turn (edit: and the current turn-in-progress).
The VP counter is implicit in the log so there's no point to showing the whole log without the VP counter. And I bet there aren't many people who want the VP counter but don't like the log.
Basically, if you like the VP counter, you are going to like the log.
If you don't like VP counter, you won't like the log.
Just two different personalities and opinions. Neither is wrong. In my opinion, the only real issue appears when trying to standardize Dominion games for a tournament or something. Do we play a more "pure" version of Dominion that doesn't include a log or VP counter? Do we acknowledge that we are indeed playing online and have these other resources available to use and it makes gameplay easier and more enjoyable for {subset of people}? That is really the only issue that I see. In my opinion, when running a tournament online, that should be one of the issues that is always discussed in the rules. For example,
the Dominion League states "By default, we play with #vpon and with random kingdoms. You can change these settings on mutual consent." If you play in the league, you abide by the rules. Could a tournament be created that explicitly states you cannot use VP counter? Sure. It wouldn't be any worse or better than another tournament with VP counter, it would just be different.