Electronic – Adjustable Voltage Regulator that operates in the uA region as a Precision Current Regulator

note: question was originally about a current limiter

The LM317 with the single series resistor between output and adjust input is actually a fixed current source, not a current limiter. You don't need the LM317 to create a current limiter, a few discrete components will do:

For a limiting at 2mA you select a 330\$\Omega\$ resistor for \$R_{SENSE}\$. If there flows 2mA through it Q2 will start to conduct and reduce the base voltage of Q1, so that its current is cut off.

edit (re changed question)
Maybe you're focusing too much on the LM317. If you need a constant current you could use the LM234 which is a programmable current source for up to 10mA. You set the current with a resistor.

Yes, you're right, but the figures you posted are the same configuration: they regulate the current over the output resistor measuring its voltage drop.

The basic difference between a voltage source and a current source, is that the first has a low output resistance (ideally 0), while the current source has a high output resistance (ideally infinite).

The voltage source with current limiting is made to provide a constant voltage in its operating range, but drops the output voltage as protection mechanism to prevent damaging the load and the source itself. Note that there are different methods of current limiting, one of which brings the current below the limit to prevent overheating.

In practice you can use a current limiting source to generate a specific current, but while a supply can handle it without problems, for an integrated devices is not a standard operating mode, and can result in wrong behavior.