I understand the confusion, a lot of terms have been conflated so frequently that the definitions of the terms themselves have been adapted to the usage rather than people getting educated on the correct use of terminology.

A good, and common example of this that was highlighted by the Simpsons, of all things, is the difference between jealousy and envy.

Jealousy is the feeling of fear of loss of a thing, and envy is the feeling of wanting what someone else has. If you look at the definition of jealousy now, it shows that jealousy basically has both definitions because the use of the word has changed because most of the population doesn’t care, and uses jealousy when they mean envy.

I’d have to dive into the etymology of the terms for evidence but on the surface and from my own knowledge, an oath is a commitment to an office, like the court, King, country, etc. A vow is a promise to an individual.

Legal proceedings have always cared about the textbook definitions of things, which is why we see oaths taken in court, when taking a position in government or the military, etc. And vows for things like marriage. I would always fall back on legal definitions for the first hint at what the etymology of the term is, and an indication of it’s correct usage, rather than the adapted usage by the public which often gets written into dictionaries.

My inner grammar enthusiast likes to use the correct terms for things like vow vs oath and jealous vs envious. It tickles my brain in a way that I like. At the same time, I try not to be so rigorous about it that I hold others to my standards. It’s more important that I understand what you mean, rather than enforcing that you use the right words when you say it.