There are a few issues going on with push starting manual transmission vehicles
As said there "may" be enough residual magnetic force left in the alternator's rotor to generate enough voltage to "wake up" the dash and computer so alternator can continue to increase electrical production to 14volts
But any "Startup Voltage" can also be absorbed by bad battery, so no dash lights would come on and no start
If you have dash lights, even dim, with key on, then you should be able to push start vehicle
Latest models may have keyless transponder, so the vehicles computer/receiver must have at least enough voltage available to recognize correct transponder code so it can at least turn on the system, which requires even more voltage, but if there are dim dash lights it should push/roll start
You could try a long rolling push start and hope alternator had enough residual magnetic force to overcome dead battery and then dash would come on once keyless transponder was recognized and then spark and fuel once computer fully boot up and saw the crank sensor timing pulse
A lot of "ifs", lol
Alternators generate a magnet field in the rotor using voltage, creates an electro-magnet, as the rotor spins the electro-magnet's field interacts with the 3 Field coils in the alternator's case and they generate a higher AC voltage which is then converted to DC
A generator uses permanent magnets in its case, and as the rotor spins, the Rotor generates electricity
So opposite of the way alternator generates electricity
But a Generator can start generating electricity just by spinning it because if has permanent magnets, it doesn't use electro-magnets like alternator
That's why the reference to "residual magnetic force" for alternator and push starting
If you continually use a piece of metal as an electro-magnet it will retain some "residual magnetism", but, in the alternator's rotors case, its also surrounded, and connected to lots of metal parts that dissipate the magnetism so wouldn't count it generating current without any voltage being put in the rotor first