Geisha is what would be called veshya in Indic, a person (especially woman) who is a professional glamour-model. (Indic: vesh = looks, attires etc ) Jap: sha = person. bei = rich person. beisha would then mean: a rich person’s woman, as are there to entertain. In short women who entertain in glamour world.
In today’s India though veshya is used derogatorily.
Note sha = shiya = shya (in Japanese transliteration) written as shyaa in Indic translit.
A truer translation in Indic in present day terms would therefore be veshyaa = mahashyaa = madam = mademoiselle (mahat-mahila)
Note: gei in Japanese means arts/artistic (hence culture), bei means rice (hence my etymo; rich ) consider this beiman might be in origin that, “the arrogance of the rich” hence in a poor society “mistrustful, traitor etc”