What's the simplest noncontroversial 2-state universal Turing machine?
This is not a real answer to your question (I don't know much about the (2,3) machine debate); but I suggest you the paper "Small Turing machines and generalized busy beaver competition". I quickly read it some time ago, and it has a nice graph with the borderlines between the 4 types of small TMs:


decidable 
open Collatz-like problem
$3x + 1$ simulation
universal




(perhaps some results have been improved).

The notion of TM used in the paper is the standard definition of TM used in papers on small universal Turing machines :

... They have a unique one-dimensional tape infinite in both directions, and a unique twoway
read–write head. There is a blank symbol denoted by 0. Initially, a
finite word, the input, is written on the tape, other cells contain the blank
symbol, the head reads the leftmost symbol of the input, and the state is the
initial state. At each step, according to the current state of the machine and
the symbol read by the head, the symbol is modified, the head moves left or
right (and cannot stay reading the same cell), and the state is modified. The
computation stops when a special halting state is reached. ...