Why is a lab PSU not suitable for driving a laser diode directly?
The section "Not an option" states:


  It is important that diode lasers always have a regulated driver in either automatic current control or automatic power control operation. A standard laboratory power supply is not suitable for driving them directly. 


You have referenced the second sentence which states that "a laboratory power supply is not suitable". The preceding sentence says that a PSU with automatic current control is suitable - but hints that it is not ideal.



Figure 1. An extract from the randomly chosen U-LD-650543A datasheet showing the power versus forward current curves at various temperatures.

We can see that, for this laser diode, that at constant current, say 15 mA, the output power will fall from about 2.5 mW to 1 mW as temperature rises from 25&deg;C to 40&deg;C. This will protect the diode at the expense of variation in output power as temperature varies so at least you'll save the laser but your application may not work reliably.

In short, a voltage-limited lab power supply would not be able to protect the laser, a current-limited supply will protect it but not give constant power and a power-regulated PSU will give the optimum performance.

Your linked article goes on to say (under "Seeking constant power"):


  Automatic power control employs a monitor diode integrated into the  laser package for feedback. Lasers with integrated monitor diodes are available in three configurations, all with the common terminal connected to their housing, which is often electrically connected to ground. The output from an integrated monitor diode is not suitable for calibration. At a given output power, the monitor current may vary by a factor of 10 from laser to laser.


So the best way to control the laser is to monitor the optical power output using a built-in photo-diode. There are three common arrangements.



Figure 2. Three different monitoring diode arrangements. LD = laser diode. MD = monitoring diode. Source: Driving Diode Lasers - A straghtforward procedure (OP's link).

The schematic of Figure 2 shows each of the three arrangements. Note that in each case the LD is forward biased and the MD is reverse biased as is normal in photo-diode applications. The voltage across the MD will increase with increasing incident light levels. The op-amp monitors this voltage and as it rises (indicating increasing laser power) it will reduce the drive to the LD transistor. The circuit will stabilise at the designed power output level.

The three options shown are:


P-type: the MD anode and LD cathode share the common terminal. The package common terminal is connected to the lower rail of a single rail supply.
N-type: the MD cathode and LD anode share the common terminal. The package common terminal is connected to the upper rail of a single rail supply.
M-type (mixed): the cathodes share a common terminal. This option requires a split-rail supply.


The important point here is that the MD is used to regulate the power to the LD.



Figure 3. The U-LD-650543A pinout.

The pinout of my randomly chosen laser diode shows that it is an N-type. Note a difference in terminology: PD, photodiode, versus MD, monitoring-diode, in Figure 2.