Well it all ultimately depends on context. As for the reason why "world of higher dimension" is not sufficient enough to warrant a higher dimensional existence is because... well also its dependent on context. I will try and give some satisfactory examples.
1. the very term "dimension" can be used within various context. And fictional works are a great example! If you search the word, you would find a bunch of stuff but ill use this as an example.
an aspect or feature of a situation.
Example: "we must focus on the cultural dimensions of the problem"
Similar: aspect, feature, element, facet, side.
So we can deduce that depending on the use case, the very term can easily be substituted with vastly different words (considering the basis of the topic)
BUT THEN HOW DO WE KNOW WHEN TO SUBSTITUTE AND WHEN TO NOT SUBSTITUTE?!
Well we do not. just take your phrase and substitute a bunch of stuff. depending on the context, some might make sense more than others. then you can at least understand the general direction the term is heading towards.
2. the particular phrase "world of higher dimension" has been used many times to represent something entirely different than a legitimate n+1 kind of increase in dimension. for example
"In the quiet overlap of a dream and the dawn, we briefly inhabit a world of higher dimension."
Now this certainly is literature at its finest. i can easily substitute dimension with depth here and it could still follow the story. so it might not be that n+1 style increase in the end.
...i hope this helps
