Another issue is that the RCC routinely accepts non-Catholic baptisms as valid, even in cases where there was absolutely no sacramental intent. This may not be what is supposed to happen but it is none the less common. Most of your priests are only concerned with water and the Trinitarian Formula. My experience is that the use of conditional baptism (common before Vatican II) has become somewhat rare today.
I suppose this is a sort of "your mileage may vary" situation ... most Catholic converts, in Parishes of which I've been a member, are received during the Easter Vigil Mass ... generally about 1/2 are Baptised conditionally, 1/4 Baptised, and 1/4 not Baptised. I'm not sure of the backgrounds of each group.