Thanks, that indeed explains most of the results.
As summary:
When adding a 50Ohm termination, i.e., a 50Ohm resistor (or rather a combination of resistors, since there is no 50Ohm one) in series with the coax output, then in series with ground, it works. Measuring across that 50ohm termination yields 4.955V. Still a slight variation, but pretty close.
And indeed, the signal generator outputs always the same voltage, but when adding a 50ohm termination (instead of a very high impedance), displaying the output signal as meant for a 50ohm load, is more practical.
So essentially you form a voltage divider with the signal generator and the 50ohm load.
When the load has a very high impedance, you still have a voltage divider, but the the signal generator's 50Ohms become negligible compared to the very high impedance, and the output is almost unaffected. There, HighZ is the right choice, as most of the voltage will drop across the high impedance load, and be very close to that of the voltage output by the signal generator.
Normal circuit rules apply: if there is another circuit in parallel/series to this 50ohm termination, the overall load may not be 50ohms anymore.
So every load circuit, DC or AC, has to be 50ohm overall, such that this voltage divider keeps working as expected. If the load circuit does not already have an impedance of 50 ohm, you need to add impedances in parallel or series to adjust.
Results depend on the cables:
Scope probes are not suitable for hooking up signals from the signal generator to a breadboard, even for DC voltage (as opposed to what was suggested in another thread, when asking about how to connect to a signal generator). Even in 1X setting! I measured mine, for the Rigol DS1054Z, and they have a resistance of 173-180 Ohms on the signal/center wire, and 0.5-1 Ohms on the Gnd wire.
A coax cable with an adapter plug and jumper wires works, but adds about another 2 Ohms. That could explain the slight variations, but still gives the best results.
Probably a coax with integrated crocodile clips or other types of clips works better. Any suggestions are welcome.