Under what conditions may I--a layman engaged in considerable private apologetic discussions--baptize those I am helping bring into the Church?
Under nearly none. Canon 861 states that the "ordinary minister of baptism is a bishop, priest, or deacon." As a layman, you aren't any of these. Moreover, you do not claim to be "deputed for this function by the local ordinary" in case the ordinary minister of baptism happens to be impeded (also from CIC 861).
Indeed, canon 861 recognizes only one condition under which you may baptize licitly: "in case of necessity." But this is not nearly as wide-open as it might sound. "Necessity" is almost always described by canonical experts as being present chiefly in cases of mortal danger. Indeed, the Roman Ritual, in providing the text to be used for baptism conferred by non-ordained ministers, opens by noting that the so-called shortened rite is to be used only in cases of "imminent danger of death."
I won't cite here the long list of other canons likely violated by private baptism outside of danger-of-death situations (though the curious might wish to consult, among others, canons 851, 853, 856, 857, and 866). Nor will I spend much time observing that, however illicit such baptisms are, they are almost certainly valid, assuming the use of water and the Trinitarian formula.
Instead, I would simply note that, according to canon 878, "the minister of baptism, whoever it is, must inform the pastor of the parish in which the baptism was administered." The reasons for such a provision are obvious, but I would not relish being the one required to report an illicit baptism to a pastor.