I think it's fine to call 移動動詞 transitive because they take a direct object. The reason why a lot of people call them intransitive is because they are 自動詞. It's appealing to want to be able to interchange transitive / intransitive and 他動詞 / 自動詞.

>that's nowhere near a "complete introduction"

I agree. There wasn't anything complete other than defining what a particle is. The article didn't even define a single particle that is classified as a 終助詞 (か, よ, な, ね, ...) despite even using them in the example sentences.

Edit:

>"genitive case"... in a language that doesn't have cases, also it's not like the article is at least consistent in calling を an "accusative case" marker

格助詞 is in JMDict as "case-marking particle (e.g. "ga", "no", "wo", "ni")." I can't be bothered to look up what cases actually are and verify this since I find applying western linguistics to Japanese to not be very useful.

I would agree if it were a direct object. I think mainstream analysis is that を is considered to not always mark the object, but sometimes the place you walk through, and as such frequently stands in opposition to に. I would argue that these are essentially adjunct phrases, not core arguments, although that's always a difficult argument to make in Japanese, since all NPs are theoretically optional, even arguments.

> I can't be bothered to look up what cases actually are and verify this since I find applying western linguistics to Japanese to not be very useful.

As usual, everyone will have their different definitions, but I consider case to be morphological (see also: https://wals.info/chapter/49). I think using vocabulary such as "genitive" muddies the waters. The genitive has a number of functions in, say, Latin or Ancient Greek, not all of which are related to possession. I think it's less confusing to call の a possessive marker - although technically, it's not always possession (e.g. 校長のスズキ先生), so maybe you could even find a better term, like "relational".

But even in English we can say things like "fly the skies" or "walk the streets", which map reasonably well to how を is used. When it's used for sentences like 自転車を盗んでいるところを捕まった (roughly "caught about to steal a bicycle") however it's harder to understand it that way. It's not the ところ that was caught, but the (implied) topic, probably "I" with no other context

This is anecdotal, but from the 2 Japanese people I've asked they see these usages of を as being the same. Ultimately the way it's classified doesn't really matter so I personally see both ways an being valid.