Just go to the tirerack site (Banner above)
and find the tires you want, go to specs for that tire, and see the diameter.
A 30" diameter tire for a 15" rim will be the same diameter as a 30" tire on a 16" rim, etc.
You can also see the section on "Plus 1", Plus 2, sizing, etc....
The aspect ratio is the relationship betwen the section width, such as 265 mm, and the sidewall height....with the lower aspect ratio having a shorter sidewall height....for the same section width.
So, yes, you can keep the same overall diameter by getting a wider section width but lower apect ratio, or, a narrower section width and a higher aspect ratio.
There is no such thing as one "Correct tire size for the rim to provide an accurate speedometer"....you can get it with lots of combinations.
You can, as described above, get any diameter tire you want on a given rim, within the obvious practical limits.
If you off road, you generally want the tallest fattest tires that can fit into the wells....to get more grip and flotation.
For on road, you want the fattest tread, but not the tallest, typically, you want the shortest diameter that you can stand comfort wise....to get the best handling.
(In your case, an easy bye would be to just get a 31x10.5/15 of course)
If you must go metric...
For any particular tire, a 70 series (aspect ratio) tire handles better than a 75 series, as the 70 series has a shorter sidewall, and will be less wishy-washy....even though the overall diameter may be the same.
I have 305/70/16's BFG AT ko's on my '01 X....they are an even 33" diameter, and have a tread width a bit larger than a 33x12.5/15's.
Some people have 285/75/16's...which are essentially like a 33" diameter 11.5" section width tire.
So - we have an example of a 305/70 being the same diameter as a 285/75....
So - as you can see, you can create any combination of width and height you need, by playing with the aspect ratios and section widths...and as long as you know what you WANT, you can find a tire that gives it to you.
I hope some of this helps.