Spool valve is a sliding cylindrical member inside a block, THere are ports on either side of neutral position. If the spool is deflected one way, it admits air to the valve actuator. If the spool is deflected to the other side of neutral, it vents air FROM the actuator. In the neutral positon it just ....bleeds.
Flapper nozzle is variable back pressure. WHen the flapper closes down on top of the nozzle the backpressure (may be connected to an output) increases. If the flapper lifts and opens, the pressure decreases. It's possible to get 3-15 psi this way, but I have never seen a flapper nozzle that will give the 0-80 psi kind of output that modern positioners have. But the designers could put a pneumatic relay under the flapper-nozzle to get any kind of output desired. Masoneilan used to use flapper-nozzles in their 2700 series pressure controller and their 8006/8012 transducer/positioner. Vibration may cause the flapper to impact on the nozzle, causing fretting and calibration shift. Masoneilan introduced a modification to avoid just that problem in 1976. ( DO I sound OLD now?)
A piezo valve uses a semiconductor piezoelectric beam that changes shape as a voltage is applied. The current draw is very tiny thus it can be energized by a 4-20 ma signal. Conversely the capacity of a piezo valve is very tiny but, like a flapper-nozzle, it sits atop a pneumatic relay/amplifier to pass the sort of flow and pressure needed. You mentioned solenoid valves: A solenoid uses a relatively greater amount of energy flowing thru a coil to change the valve position magnetically. The only valve positioner I know of with solenoid valves is the Leslie DVC. Because of the energy requirement of the solenoids it must be a 4-wire device, not loop powered.
Fail freeze is not exactly related to zero-bleed, but one is necessary for the other. The fail-freeze option causes the piezo valves to lock closed if the signal goes below 4 ma. (user-assignable value, but 4 ma is common). NORMAL operation of a positioner is for the positioner to interpret anything less than 4 ma as ZERO and to vent the actuator. SO if you give a failfreeze positioner 4 ma the valve closes. But if you are running along midscale and the wire gets cut, the valve stays in last position. Siemens is not the only failfreeze positoner. PMV has it on the D3 as an option. There may be others.