The person doesn't need to have an idea of the theological jargon but will agree to the spiritual truths that the jargon represents however it is expressed.
For instance, the convert may have never heard of the term limited atonement or particular redemption but, when they are confronted with the notion that Christ's death actually paid for certain sins for certain people (chosen before time), I don't think a born again person would necessarily disagree with that.
Well, there are those that say Christ's death actually paid for all sins for all people. I think this idea is taught in the bible all over the place.
It's the qualifiers I disagree with: certain sins, certain people. Does that mean you do not consider me a "born again person?"
There's also loads of Christians that reject that "Chosen before time" means chosen Some instead of Others.
It seems to be a whole lot more qualifiers to being a believer than Jesus or the Apostles mentioned. You've got a pretty small list, here. Pretty much only reformed churches/believers get a pass.