I think the TX 183 Toll Road in Cedar Park and Leander should be part of mainline US 183, and existing 183 should be either Business 183 or a local street (N./S. Bell Rd.). Was the Bergstrom Expressway given the TX 183 Toll designation because Texas's Toll Roads have either a state highway designation, or are otherwise unnumbered?
That's a little complicated. The 183A toll road was originally planned to be built by TxDOT and designated as US 183A. When the regional mobility authority got the road, it was not made into a state highway, but it kept the 183A name, which is just a name, like "Dallas North Tollway." The 290 and 183 toll roads are the same, they're just names, not highway designations. TxDOT's toll roads have highway numbers. Near Tyler, the road planned as Loop 49 was built by the regional mobility authority and the loop designation was removed. People still call it Loop 49, but it's officially Toll 49, which is the name given to it by the RMA. The 360 Tollway in south Arlington is the same, just a name, with TX 360 running along the frontage roads. Other NTTA roads have non-numerical names. The Sam Rayburn Tollway was originally planned as a state highway owned by TxDOT and leased to a private company. It would have had the SH 121 designation. When NTTA won the project, it was removed from the state highway system.
In general, a TxDOT road has a state highway designation and a road that is owned by a local or regional authority does not. What I believe an exception is I-169. I think that is owned by the RMA, but it is designated on the state highway system. There may be some other arrangement. It may be owned by and leased from TxDOT, like the southern half of TX 130. Or it may be that TxDOT owns the road and the RMA just handles the billing.