If by “a regulated power supply” you mean specifically a non-switching one as your second paragraph, then “a regulated power supply” and “an AC/DC adapter (linear, non-switching)” are exactly the same thing.
There are basically only three categories of Things You Plug Into The Wall And Get DC Out:
Unregulated linear power supplies. These contain a transformer, a rectifier, and smoothing capacitors. Their voltage output is significantly higher than rated when less than their rated load is applied, and their output has lots of “ripple” (a bit of 120 Hz AC on top of the DC output).
Regulated linear power supplies. These add a linear regulator circuit to the above, producing a stable, lower, output voltage. The difference in voltage, times the output current, is dissipated as heat; therefore they are most efficient when under maximum or minimum load.
(Historically, an electronic device might use an external unregulated linear supply and place the regulator on the main circuit board instead.)
“Laboratory” linear power supplies may add a current-limiting feature to this design. This is an addition to the control section but the power-handling components are the same.
Switch-mode power supplies. These produce a stable output voltage (or current, as in LED drivers) and do not dissipate heat, but they can produce RFI due to the high switching frequency. A well-designed switching supply will keep the noise inside the box by way of shielding and output filtering capacitors.
Any of these may come in “wall-wart”, “lump on the cable”, or desktop/benchtop form factors. This is determined more by their power rating (and hence their necessary physical size) than anything about their type of functionality, though a linear power supply will be larger and heavier than a switch-mode power supply of the same capabilities.