The original circuit is a "low-side" current sense. The first variant is a voltage source with a high-side current sense. The second variant is the same but is now adjustable (tweak trip current). The third variant replaces the two diodes with a transistor to make it a little more precise. All these circuits are not going to be very precise as the Vbe junction of any transistor is sloppy and changes a lot of temperature. The low side vs high side just depends on your application (maybe you don't have access to the high-side).
The three transistor version is going to be the most precise and give you adjustments to trim some error and tune the current trip level.
The LM358 OpAmp is current limited to typically 40mA.
With a +-5V supply you are limited to +3.5V and -5V output excursion.
If you short the output you will source 3.5/50=70mA or sink -5/50=-100mA into the grounded 50Ohm series resistance. This will result in a max of 0.245W (when positive) or 0.5W (when negative) dissipation.
Unless you have a constant -5V output that is shorted you will see an average that is less than 0.5W dissipated, if you remain in the linear portion of the output (+3.5V -3.5V) you will have less than 0.25W into a short. You do not need extra short circuit protection at those supply voltages.
The OpAmp is rated continuous short circuit proof with less than 15V supply.
If you have it calibrated for 2V open circuit you will see 1V across a 50 Ohm load and have a matched source impedance and not be able to approach any device limits.
I would suggest fast diodes connected from the output to the supply rails to prevent external devices from causing device limits to be exceeded.
EDIT:
The max current limits calculated cannot be achieved with this particular OpAmp as it lists a short circuit output current of 40mA typical 60mA max and safe for continuous short circuit, it is inherently protected and the current limit protects the output resistor. Higher output currents could be reached with some other types. The max output voltage is listed as the positive supply - 1.5V hence the 3.5V positive limit with a +5V supply, devices that can swing closer to the supply rails are also available and have their uses.
All of the numbers used are available in the data sheet, either in the text, tables or graphs. Note 1 on Table 1 warns about short circuit dissipation limits with supplies over 15V
Calibration at 2V was just my suggestion for the choice of gain components so full-scale digital output would be calculated to give 2V output or 1V into a matched 50 Ohm load, these low voltages would also be self protected as the currents would be even less and within the 30mA supply limits to maintain accurate operation.
I keep forgetting to point out that the possible supply current limits of 30mA would be reached before the resistor dissipation or OpAmp limits. This may cause unexpected behaviour especially if one supply rail were to be reduced more than the other for instance, though with this device this is less likely as it is a tracking dual regulator. It is possible to configure the Mitsubishi M5290P for more than 30mA with external transistors so it is not certain that is will be current limiting at 30mA in this re-purposed power supply circuit.
Best Answer
The 2N2222 can dissipate a maximum power of 0.5 watts so your biggest problem is that when the motor is stalled, the transistor is going to be dissipating over 2 watts of power and will rapidly expire.
So, choose a more powerful device for T1 and you will find that the resistor you ask about can be in the range of a few hundred ohms to maybe a kohm.
New section about R1
If you study the NPN BJT as an "emitter follower" (common collector) you will discover that the voltage on the base will be about 0.7 volts above the voltage on the emitter. It can't be much more (maybe a volt max) because there is a forward biased diode between base and emitter. It can't be much less else the transistor isn't being turned on very much so, generally speaking the goldilocks number is 0.7 volts. OK so far...
Now if your supply voltage is 24 volts and you need to provide (say) 23.3 volts at the emitter for the "load", the base has to be at (or about) 24 volts. But herein lies the problem because, to control the base voltage with the 2nd transistor (in order to control current), you need a resistor between 24 volts and base. This creates an extra volt drop because of the base current needed to switch on the BJT. OK so far?
The load is about 120 mA and operating the BJT close to saturation might mean a gain as low as 20 hence, the base current needed is 6 mA. But, T2 doesn't want to control T1 with a small value of R1 because it might have to become a power transistor like T1 so, it's a compromise. R1 is chosen to drop maybe a volt at 6 mA which yields a value of 167 ohms.
The down side of this is that now, the emitter can only be raised as high as 22.3 volts on a 24 volt supply but, if you can live with that then all is good.
MOSFETs have other problems that can make life hard reaching a source voltage as high as the emitter voltage of a BJT but, things are made easier in the gate draws no appreciable static current hence R1 can be 10 kohm. Making it too high can cause problems in that the current limiting takes an appreciable time to kick-in.
It's a simple circuit but full ov subtle surprises.