The words "monk" and "priest" do not really work as good translations of the Japanese terms, and were picked, obviously, from a Judeo-Christian vocabulary. "Priest" carries the feeling of working some power to intervene with God/the spirits, and most Zen "monks" only reside in monasteries for short periods as part of their training ... so both words are not good fits (except when the person is actually residing in a monastery and might be described then as a "monk".). The best translations might be "Companion" "Guide" "Teacher" or (my favorite) "Rabbi (which also means "teacher")".
A very nice old term for a teacher used in China is "shanzhishi" = a good spiritual friend (Sanskrit kalyanamitra.)
I prefer "Zen clergy or teacher or minister ". One of the many Japanese terms usually (and awkwardly) translated as "monk/priest" in English is actually closer to "Buddhist companion" , which I care for very much ...
?? (the first kanji derives from the "san" of Sanskrit sangha = community, and the second means companion)
So "Buddhist companion" or "Sangha Friend and Companion" may be the most accurate.
Of course, many "Zen priests" in Japan and China do reside in temples in which they are largely concerned with performing funeral and other ceremonies for parishioners to appease the spirits, bring good fortune or the like. In such case, "priest" is not inaccurate to describe such folks.