Why does copied MySQL database have different data than source?
Honestly, I expected you to say MyISAM since I've never seen InnoDB generate a backup that seems inconsistent with what you think is in the table.

As a DBA, I don't trust my data to anything that I can't watch while it's working and peer into its intermediate stages, so I rarely use GUI tools.  My approach to this issue would be to use mysqldump to extract the data into a file and then review that file manually.  Satisfied with the sanity of it, I would load it onto the target server with the mysql command line client and see what the results are.

The specific option I'd suggest with mysqldump in this case would be --skip-extended-insert, which generates one insert statement for each row row in each table.  This is in contrast to the normal format, which combines groups of rows into a smaller number of insert statements as an optimization for faster restoring... a file created with this option will restore somewhat more slowly than one created without it, but the tradeoff is that the files are substantially easier to parse with your eyeballs and see what the data looks like during this intermediate stage of the process.  

Since my bet on MyISAM didn't pan out so well, I'm really unsure whether to put money on the dump file looking right and some kind of strangeness on the target server to scramble it up on the way in... or for the dump file to come out looking wrong, though I really will be surprised if the dump file comes out looking wrong... but then again, if the dump file looks right, I'd be pretty surprised if it doesn't restore correctly. 

After reviewing the file, if the content looks right (i.e., find the line in the file that represents the 438/2255 row you mentioned above, and see which version it represents... and make sure there's only one such line in the dump file) load it onto the target server and see what the results look like.  If the contents of the dump file don't look right, then of course, restoring it would be unnecessary and we need to look at the origin server for something strange.