reviveing a bitcoin core wallet
Bitcoin Core 0.10 was not released until 2015. It is impossible for you to have used that version in 2013 as it did not exist.

However the wallet format itself has not changed since at least 2012 (and earlier really) so it is completely fine to upgrade your software. The wallet.dat file can still be used in Bitcoin Core 0.15. All you need to do is make sure that Bitcoin Core is shut down, go to the Bitcoin Core data directory, and copy the wallet.dat file from there to the Bitcoin Core data directory on your other machine.