In brief, it may make sense to think of the choice of preposition as both based in dialect and based on semantic requirements.
First, I agree with your general findings that the choice of preposition in this context is based in dialect. Other informal sources attest it, like several people in a Word Reference forum post. UK dictionaries also tend to model the open ... at usage: Oxford Dictionaries (at), Oxford Learner's Dictionary (at). This pattern seems to be common enough that many American and UK speakers don't recognize it as a dialect difference until they encounter the other version. It's that subtle.
That said, there are also non-dialect explanations for the choice of preposition. Famed British linguist David Crystal wrote a blog post in 2011 explaining the difference semantically, though his choices (perhaps writing from a UK perspective) are "at" and "on":
A correspondent writes to ask if he can say both ‘Open your book on page...‘ and ‘Open your book at page...’ Is there a difference?
...
‘Opening a book’ is an interesting example of overlap between the two perspectives. In one way it’s a reference to location - so, ‘at’. Most people would open a book ‘at’ a particular page. But people have a semantic reason for asking someone to open a book at a particular point - so ‘on’ isn’t ruled out. In the first case, they’re thinking ‘where’; in the second, they’re thinking ‘what’.
If Crystal were writing from an American perspective, "to" would have a sense of location or position with "open a book," but "at" or "on" would be used in other cases. Perhaps that sense of to would be as you describe in (A), as the preposition in the phrasal verb "go to," or as something like definition 4a for the preposition "to" in the Oxford English Dictionary:
Expressing simple position: At, in (a place, also figurative a condition, etc.).
So in American dialects, there may be a further semantic breakdown between "at" as denoting a static location, "to" denoting a purposeful action toward a location, and "on" as a more literal descriptor of place.
In other words, using Crystal's language, at denotes "where (static)," to denotes "where (purpose or goal)," and on denotes "what." Because UK English already uses "open at," within that system at denotes both kinds of location ("where") without a further distinction except in a phrasal verb where to is already baked in, like "go to."
Best Answer
The Chicago Manual of Style does not include the word "teach" in their list of common problematic words and the correct preposition construed with them. It does include the verb "instill," and indicates that this verb is correctly used with the preposition "in."
More interesting, though, is the inclusion of the word "based." It indicates that the following adjective should be "on" when used preceding a premise, and "in" when used preceding a field of study.
With this guideline, I would assume that the preposition used should be selected to convey either a summary of the class subject ("on"), or an immersion in the class subject ("in"). For instance, I would take a class in standard deviations if I was pursuing a degree in mathematics and had already acquired any prerequisite credits. Or, I might take a class on carpet-laying at my local community college; the class subject is limited and does not imply a broader range of study.
It stands to reason that common use may default to prepositional accuracy without much conscious thought on the part of a speaker. The preposition is likely considered further in written communication to convey credibility.