monk: /məNGk/ Noun
A member of a religious community of men typically living under vows of poverty, chastity, and obedience.
Catholic priests are monks by that definition. A monk leaving the monastery and removing himself from the physical presence of the community does not lose his monkhood. And seminary school qualifies as the community.
Also, you say I'm conflating celibacy and being unwedded. I argue that in Western Catholic teaching tradition, they are one and the same. If you are unwed, you are expected to abstain from sexual relations. If you are wed, you are to follow God's commandments and propagate the human race. I spent 13 years as a good little Catholic learning that.