Judging from the second error, you failed to properly upload at least four different files, and possibly the updated contents of the Modules directory... I would advise going through the zip one more time and ensuring that the files on the server are indeed in sync with what you meant to upload. Sometimes FTP servers silently refuse to overwrite files - you may need to carefully delete the files on the server before uploading new copies.
(Hitting uncached pages on your board reveals the old version number, so you're definately missing files...)
From there, the upgrade procedure is indeed one button... log in, hit the button, then sit back and wait as it tries to do everything for you. Among the things that the upgrade button does includes making sure all of the expected file versions are present... of course, if you can't even log in to run the thing in the first place, that's rather useless. There has to be a way to fix that... %)
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