Does this effect depend on anything else than the JFET/BJT input
stage?
Slewing can happen in any stage of an OpAmp, it can happen in the input stage, the output stage and any of the intermediary stages. It occurs whenever a capacitor is driven by a fixed current source. For a given configuration one stage sets the limit and determines the slew-rate often this is in fact the input stage.
A typical input stage consists of a differential pair with a tail current source. In equilibrium the current of the tail current source splits equally and when driven hard on transistor takes the whole current. The required voltage to turn one of the transistors (almost) off determines the onset of slewing. It's a fixed voltage for BJTs and a variable voltage for FETs.
If slewing happens in the output stage and the output stage is not symmetrical (e.g. class A) it is possible to have different slew-rates for falling and rising edges.
Are there input stages where an even lower differential input voltage
leads to the maximum slew rate? Perhaps some sort of hybrid?
I don't know of any off-the-shelf devices that do this, but certainly there are adaptively biased OpAmps that increase the current through the input stage to improve the slewing behavior.
Even if there are only two types, do different op-amps of the same
type (say, BJT) have different levels?
BJTs have a fixed level, unless emitter degeneration is used and FETs can have different levels.
There are several contributing factors to the poor performance of the circuit:
1. Insufficient supply voltage
Minimum Supply Voltage for the TL072 is listed as 10V. Using only 5V is well below this minimum.
2. Insufficient supply voltage headroom
The TL072 is not able to drive its output up/down to either voltage rail so expecting it to output 1V under Vcc is not realistic. See this question for more details.
3. Improper component selection
You should be using a comparator in this circuit and not an opamp. While the diagrams and basic function of an opamp and comparator are similar they are not substitutes. See this question for more information.
Best Answer
Generally an op-amp's slew rate is a limitation of the op-amp in being able to change it's output in one direction or the other for a step input change. It isn't affected by gain but it can be easier to mistake reduced bandwidth for slew rate limitation when measuring with a significant gain.
Remember that in pretty much all normal op-amps gain x bandwidth is largely a constant value so trying to measure slew rate with a big gain can lead to a false measurement if you are not careful.