Power supplies are today almost always switched mode power supplies. That's because it's the only possible way to create a power supply with stable voltage as a function of load, low cost, low weight, high efficiency, perfect power factor (sinusoidal current input) and ripple-free output.
Old transformer-based power supplies almost always had huge variation of voltage as a function of load, and if they didn't they just used a linear regulator to dump the excess voltage as heat, becoming extremely inefficient. They weighed a lot for a very low current capability. The power factor was crap due to large capacitor current surges, and if they didn't have a linear regulator then the capacitor ripple was huge.
A switched mode power supply converting AC to low-voltage DC can be for example forward converter or flyback converter.
They create the required voltage by rapidly switching current through an inductor via a MOSFET.
In a forward converter, current through the output inductor is going through the load. At any time, the controller has the option to also supply more current to the inductor via a high-frequency switched-DC transformer.
In a flyback converter, current is alternately flowing from input to inductor (energizing it), or from inductor to load (de-energizing it). The inductor has two isolated windings.
By adjusting the duty cycle, it's possible to adjust the current and voltage through the load. This is the basis of voltage and current limits at most lab power supplies.
To limit voltage, you need to measure it -- nothing else, the rest is in the control algorithm. Same for current, to limit it, you need to measure it, and the rest is in the control algorithm.
It's probably very hard to find a switched mode power supply with absolutely no form of overcurrent protection. It's so cheap to add there that it's almost always there.
Also, do note that short circuiting a SMPS lab power supply isn't advisable. For example, I have a 30V 10A power supply. I have tested short-circuiting it at voltages of below 10V, and at few amperes of current. It creates a magnificent spark show when you remove the short-circuited leads. Not only that, but the metal in the short-circuited leads is damaged, because essentially what you're doing is using the lab power supply as a welding power supply.
I suspect that at 30 volts, and 10 amperes, the spark show would be so huge that you have to wear eye protection or else some of the sparks can fly into your eye.
You don't want to destroy your power supply leads by creating spot welds into them! I have, and won't do it again.
Of course by short circuiting before turning the AC side on, adjusting the current, and then turning the AC side off without removing the leads, and only then removing the leads you can eliminate the spot welds and the spark show.