It is odd how they figure this stuff, especially when you have metric stuff, the older "inch" stuff.
Some tire do have the full spec for, ie your 23x8.5-12. As you have found out, it is 23"diameter, 8.5" wide, and uses a 12" rim.
For those 12.4-16's, that is an older way to spec it. It is a 12.4" effective tread, with a 16" rim. They do not show the diameter in this spec. In the old days a lot of car/pickup tires were specced that way. I remember my Dad's Jeep in the early 70's running a 10.50-15 tire.
The most informative way to find tire info is the books, or manufacture web pages. It is most important for these 4x4 CUTs which require front to rear ratio matching, since the front tire are shorter than the rear.