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.
Types of power supplies
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 less efficient than an unregulated supply, and most efficient when under maximum or minimum load.
“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.
The “Class 2” mentioned in your question title is a safety standard: it says that the adapter is guaranteed not to produce too much voltage or current, so that the output will not pose the kinds of safety hazard AC line power does. This means both under normal operation (the maximum output current must not be too high) and possible faults (internal physical protection against the AC input jumping over to the output side).
Replacing power supplies
Historically, an electronic device might use an external unregulated linear supply and place the linear regulator on the main circuit board instead. Modern devices may also have linear regulators or switching converters built in, in order to accept variable voltage or to produce multiple lower voltages for internal use. In this case, you could safely replace a switch-mode power supply with an unregulated linear one, as long as its open-circuit voltage is not too high for the device's input circuit components. This will reduce RFI but increase waste heat.
It is safeOK to replace a regulated linear power supply with a switch-mode power supply of the same output voltage and current rating, or vice versa. The tradeoff you are making is efficiency versus RFI.
It is not safeOK to use an unregulated power supply with an electronic device that does not have an internal regulator, as the varying input voltage may be too high unloaded or too low unloaded, causing damage or incorrect operation. It may be safeOK to use an unregulated power supply with a merely “electrical” low-voltage device having only lamps, motors, and such, but it should be chosen to match the current rating of the power supply to the actual load of the device.