In aircraft that use a "split switch," you're talking about two different switches. That the switches are the same color and close to each other is the only functional commonality. Some are mechanically interlinked such that one may be turned off without the other, but the other is turned off, they both go off.
Use the proceedures recommended or directed in the approved data you've received. In most cases that's your aircraft flight manual or pilot operating handbook. These proceedures may vary from manufacturer to manufacturer, and will certainly be aircraft specific.
Generally speaking, you should leave the electrical generator (alternator or generator) off during engine start. This is true weather it's an integral starter generator, or separate starters and generators.
Turning on the alternator or generator during the start means that a certain operating resistance is had by the gen/alt. This means youre engine is carrying a bigger load, or meeting more resistance during the start. Your aircraft battery, if used as the electrical source for the engine start, is working harder; more current and a bigger voltage drop occurs.
Leave off the gen/alt to make life easier on your battery and starter motor.
During engine start, with the gen/alt on, your battery acts to "absorb" transient voltages, but it can't protect equipment on line. Voltage will rise and fall, and transient voltages may exceed the ability of the protection equipment (eg, regulator) to control during the start process. Accordingly, a potential exists to damage equipment on line.
Leave off the gen/alt to protect equipment on any affected bus. Avionics, radar, etc, are the most susceptible to voltage spikes.
Further, as mentioned before, initiating field current after start enables a positive check at operating speed for the gen/alt for voltage and current flow.
Leave off the gen/alt until a stable start has been achieved for a better positive check of the gen/alt and electrical systems.
Above all, follow the proceedures provided by the manufacturer for your aircraft, always.