Cave Man HTML
I fixed my SiteMeter account, using cave man "programming": delete all the HTML around the faulty SiteMeter counter stuff in the template, and then reinstall from scratch using the old account info.
Has to be a better way, but that will certainly work.
I had tried getting SiteMeter to overwrite the code, or just change the account associated with the meter, but it wouldn't do it. It would SAY it was done, and that the changes were made, but they wouldn't show up.