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 of Western missionaries in the 19th century. "Priest" carries the feeling of working some power to intervene with God/the spirits, and most Zen "monks" now only reside in monasteries maintaining celibacy 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". 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.)
In my view, the best translations might be "Companion" "Guide" "Teacher" or (my favorite) "Rabbi (which also means "Teacher")".
A very nice old term for a Buddhist teacher used in China is "shanzhishi" = a "good wise friend" (善知識, Sanskrit kalyanamitra.)
I often use "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 ... 僧侶 (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.