There was no Bitcoin Core in 2010. The first Bitcoin Core release was in March 2014 (0.9.0). Before that there was Bitcoin-Qt (with first release in November 2011), and (experimental) Bitcoin (with first release in January 2009).
It was only the name that was changed and there isn't any big difference between them. Bitcoin QT was actually rebranded to bitcoin core in that time. Whether a wallet has been created by bitcoin QT or Bitcoin core, it should have created a wallet.dat file.
Please correct me, if I am wrong please.
To OP:
Since you have used your wallet for mining in 2010, you should have a wallet.dat file.
As already mentioned above, try to find a wallet.dat file or your private keys.
Note that if your wallet.dat file is encrypted, you will need the passphrase for decryption as well.