What would you call this power supply topology?
I have seen the triac PSU circuit before. I have heard it called "Primary side phase control". It is old school and has a bad peaky input current waveform and lots of output ripple. The power factor and output ripple on the primary side phase control PSU is worse than an uncontrolled diode rectifier.

Where I have encountered it is on battery chargers manufactured in a time where output ripple was not a concern and nobody worried about power factor. In most applications the desired output voltage is much lower than the incoming mains. This means that a cheap TO220 triac can be used for a say 1KW or more charger. The more orthodox secondary side phase control would need 2 SCRs rated at the load current with a center tapped secondary. In the 1970s when power semiconductors were very expensive or unobtanium the triac circuit was tempting. Transformer magnetising inductance and leakage inductance are important transformer parameters that you have to get right if you expect reliability.