I don't work with black holes, but I've been in the field of astrophysics for several years now, and until I read this question, I had never heard of either the two terms.
But a query on NASA/ADS, the primary database for "professional" astronomical papers (both refereed and non-refereed) yields 54 hits for "peribothron" vs. only 10 hits for "perinigricon". Not a lot.
As your links reveal, the etymological origin of peribothron/-nigricon is "near the black hole" in Greek and Latin, respectively. The reason has been to conform with terms such as perihelion, meaning "near the Sun". The first to use the Greek version in the literature was Frank & Rees (1976) who write in a footnote:
$^\star$We are grateful to W. R. Stoeger for suggesting this word, derived from the Greek bothros, a pit.
William Stoeger was the second author Martin Rees' student (and Stephen Hawking's classmate).
The first to use the Latin version seems to have been Schödel et al. (2002), but they don't offer any explanation for the term.