The last one is most correct. There are differing opinions, but my graduate Latin professor who is very well regarded scholar of late antique Christianity, especially the age of Constantine, told me that it wasn't uncommon for people to be baptized late in life. This especially applies to rulers.

/u/Enrico_Dandolo is correct; the last is the most likely explanation. The effects of baptism, as rationalized by many in the fourth century, was a one-use-only kind of deal. The spiritual cleansing of baptism was a grace that could only be received once, and for this reason, many decided to delay the event (some, like Constantine, until the last possible moment). While the whole scenario might seem a little opportunistic to us – though contemporaries such as Augustine and Ambrose certainly railed against ‘perpetual’ catechumens—it is still reflective of the reverence and seriousness with which late ancients treated the sacrament. To undergo the sacrament of baptism was to commit to a pretty serious life change, and any further sins would require rigorous repentance to expunge.