score:5
From what I gathered, without knowing which book it was you were reading, both towns appear to have been two mining centers, established by the mining company El Boleo, and they must have just taken their names from existing landmarks (mountains or streams β there's a Purgatorio stream near Santa RosalΓa, in the center of that mining area). In any case, looks like it was nothing strange for the Spaniards to name places like that, for, from what I have seen, there's at least one Purgatorio stream in Sonora, another one in Jalisco and, much farther away, one in Colombia. The Purgatoire river, being in Colorado, it's my guess that it was so named by the Spaniards and got its name anglicised (Protestants don't believe in Purgatory and it would be odd for them to name anything after it).
Providence, Rhode Island, was named, as also was every Spanish town of that name, after the Divine Providence by its founders, deeply religious people, so in that case it was just coincidence.