I tried it before on my Cyber Power 810w UPS and the Sylvania 4.5w LED filament lamp got little bit bright while I unplug the UPS from the wall. Not sure if I use it for long periods of time, because I'm aware that uninterruptable power supplies output a simulated wave and not a pure sine wave (although some now have pure sine wave output) when its on battery.

Is this safe?

It depends how the ballast is constructed.
If it uses capacitive dropper, it is not ok.
Others, so a switch mode DCDC or a linear CCR based are OK (they both start with the rectifier).

Quite a reliable way to say is to try to supply it behind a bridge rectifier.
If it glows normally at full power, it uses ballast compatible with the rectangularish "modified sinewave" of those UPSes.
If it only flashes at power ON (or not at all) and then glows very dimly, there is very high chance it uses the capacitive dropper circuit, so incompatible with the MSW.
There could be some oddball topology that behaves on this test like the capa-dropper but actually be OK with the MSW, but I would not rely on that...

It depends how the ballast is constructed.
If it uses capacitive dropper, it is not ok.
Others, so a switch mode DCDC or a linear CCR based are OK (they both start with the rectifier).

Quite a reliable way to say is to try to supply it behind a bridge rectifier.
If it glows normally at full power, it uses ballast compatible with the rectangularish "modified sinewave" of those UPSes.
If it only flashes at power ON (or not at all) and then glows very dimly, there is very high chance it uses the capacitive dropper circuit, so incompatible with the MSW.
There could be some oddball topology that behaves on this test like the capa-dropper but actually be OK with the MSW, but I would not rely on that...

Which one of these ballast implementations is the most common one in retrofit (e27) led lamps?

The lamps that have no place for electrolitic capacitor such as filaments - Use something that goes off 100 times/sec. This may be some constant current driver or a driver that is supposed to cut in and out at certain points of the sine wave. Running it on a different wave shape may make it work at a different duty cycle (or energy per cycle, i.e. integral of the power over the cycle) than designed

The lamps that have a neck that could contain an electrolitic capacitor - They can be anything. With capacitor it's mostly the peak of the voltage waveform that matters, and the quality of the feedback in the PS (ranging from none to fairly good). Ideally the peak of the "modified sine" from the UPS would be lower than that of a sine (of same RMS value), but the modified sine output from the UPS is not perfect and it may have overshoots, especially when the load on the UPS is the few Watts LED lamp alone

The cheapest lamps from Ebay/Aliexpress and some of the wackier lamps in general (corns etc) are the ones where series capacitor drivers can be found (but they still can have all other driver types too)