I don't know that our universe isn't a subspace of some higher-dimensional structure. That has nothing to do with what the specific word "dimension" means and has always meant outside of sci-fi. There has never been a sound reason to use the word "dimension" to describe the other possibilities you mention. If you want to write a technical document about those possibilities and be taken seriously, you need to find a better term. Invent one if necessary.
In that case I think it's you who is misunderstanding the word. Unfortunately I haven't the time to give you a course on Linear Algebra so unless you can explain in more detail why you believe that the word "dimension" is inappropriate for the case I described I'm not going to be able to help you.
Linear algebra is exactly where "dimension" gets its abstract but very limited definition as a property of spaces, best summarized as an axis or direction, rather than a kind of space itself, which is what you need for anything like an alternate universe. I know this because I have in fact taken a course in it. I'm not sure what point you think you're making at this point.
> Linear algebra is exactly where "dimension" gets its abstract but very limited definition as a property of spaces, best summarized as an axis or direction, rather than a kind of space itself
Yes, but adding an extra dimension to a given space (more correctly stated as asserting that a space of dimension d is embedded in a space of dimension d + 1) vastly expands the "number" of points that are available, so adding a dimension doesn't just give you access to an additional d-space but to a whole continuum of other d-spaces. I think this is what people are referring to when they use the word "dimension" in this kind of context.
Most people have no such precise idea when they talk about "dimensions". Those who do would, when pressed, admit that even if you could be said to travel through or along some other dimension to reach another world, or even an infinity of other worlds, said dimension is not the same as those worlds, just a term for a relationship between them.
Moreover, I don't care about what people think they're saying with their crap terminology. I demand actual accuracy, or at minimum precisely delimited uncertainty, from people taking millions of dollars in government funding to write about aliens. This is not too much to ask. (You do remember the context, right? Evaluating the credibility of a purportedly factual report written under a government contract, not a bar debate about cosmology)
