Source: It’s the terminology banks use when discussing paychecks. If you get paid on the same days every month (like the 1st and the 15th) then you get paid bimonthly. If you get paid every two weeks regardless of the date, (like every other Friday) then you get paid semiweekly.
Interesting, I've come to understand those prefixes to have the opposite meaning. I looked into it, and apparently the actual usage of bi vs semi is not super consistent, and has been historically ambiguous and especially difficult because of how confidently people tend to believe others share their understanding on the words.
You'll notice a lot of the words you listed include both "twice per" and "once every other" in their dictionary definitions. Of course, dictionaries are descriptive and not prescriptive indicators of language, and I think it's fine to argue one's case to keep the terminology consistent at least within the bounds of a conversation.
I guess the takeaway is that it's good to have a context under which the terms are consistent, like with banking, but also important to understand that there are other contexts with definitions that differ, and that those different contexts aren't incorrect.