Why does the TUF formula use \$V_{rms}=\frac{V_m}{\sqrt{2}}\$ and not \$V_{rms}=\frac{V_m}{2}\$ for a half wave rectifier
I think I got it and it is quite silly. There are two \$V_{rms}\$ voltages, it can be considered as:

\$V_{rms(input)}\$ : The voltage that is measured before the diode i.e. at the input of the diode and it's the direct output of the transformer.

\$V_{rms(output)}\$ : The voltage that is measured after the diode.

Also, since it's a diode in series, the current before the diode is the same current that is after the diode hence there is only one \$I_{rms}\$

The AC rated power uses the \$V_{rms(input)}\$ which is quite obvious because that's the voltage across the secondary winding of the transformer.