Potentiometer controlled current regulation
It has two leads and the resistance between them is 0.9 Ohm. Assuming the voltage would be constant, max current it can draw would be 13,3 Amps. 


Correct.


  I need to be able to control how far the plunger can get pushed out. I am led to believe that with constant voltage this can be achieved by varying (limiting?) current to the solenoid.


Slightly garbled. Voltage and current are related as dictated by Ohm's Law. If you can reduce either one the other will reduce too.


  Is it doable with LM317 or something similar (LM350?)? 


13 A is high current and any linear regulator will have to dissipate all the power dropped in the regulator. e.g. If the current is 10 A at around 9 V out then there will be 3 V across the regulator at 10 A so \$ P = VI = 3 \cdot 10 = 30~W \$ heat dissipation requiring a large heatsink.


  Would a thyristor or a transistor be of any use? 


Thyristors are good for AC. Once triggered they remain on until the current through them falls to zero so in your circuit the thyristor would remain on until something else switched it off. You might as well use the something else to switch it on as well.

If, however, you wish to control from a transformer a triac (a bi-directional thyristor) control circuit is usually quite simple. Power is adjusted by delaying the turn-on of the AC waveform.



Figure 1. Varying the turn-on point of the triac allows variation of voltage or current from almost 100% to 0%.

Note that highly inductive loads such as solenoids require special consideration for reliable operation.


  Is there even a proper 'name' to the kind of circuit I am looking for?


Read up on pulse-width modulation (PWM). This technique gives pulses of energy to the device being controlled (motor, lamp, LED, etc.) with the pulse width determining the average voltage / current delivered. The pulse frequency is chosen to be high enough that the device operates smoothly. e.g., the motor doesn't vibrate or the lamps don't noticeably flicker.



Figure 2. PWM power control showing 75% - 25% - 75% - 0% power sequence.


  I know current limiting can be achieved by increasing resistance, like with a potentiometer, but the pot might become dangerously hot, right?


Correct. High powered "rheostats" were used in theatre lighting, electric tram traction control and probably by Flash Gordon but have mostly been replaced by electronic control due to increased efficiency.