Am I Catholic if I received private instruction instead of going through RCIA?

You are Catholic

You were validly received into the Church.

The purpose of RCIA

RCIA is primarily meant to catechize and sacramentally initiate the unbaptized. Many parishes, for practical reasons, put catechumens (the unbaptized), candidates (baptized non-Catholics) and adult Catholics seeking confirmation in the same program. That can be done, but private instruction for a baptized Christian better fits RCIA's purpose.