Is the 'Isha prayer that I offered for the second time valid, knowing that I had previously prayed it as two rak'ahs, thinking it was 'Isha, and then completed it to four?

The first 'Isha prayer is valid if it occurs after the entry of its time, and the second one becomes a supererogatory (nafilah) prayer. If the first prayer occurs before the entry of its time, it is not valid as an obligatory (fard) prayer but becomes a supererogatory one, and the prayer performed with the congregation then becomes the obligatory prayer.