It's a complicated question.
The DI out from an amp is normally pre-amp out, that can be post or pre-EQ. An external DI might be post power amp where the DI is connected parallel to the speaker cabinet. The advantage of this is that there's a contribution from the power amp. Of course, a DI doesn't need to be through an amp at all.
Mic'ing a cabinet can be accomplished in different ways. The sound of the cab contributes to the tone. The speaker can be close mic'ed. The position of the speaker matters. This can make the sound brighter or more bassy. The mic is important. It makes contribution as well., every mic has a characteristic frequency response. The cab can also be mic'ed from a distance, for instance a meter away. This allows the mic to pick up room ambience. I like the sound of some air between the speaker and the cabinet.
Tone can be contoured by the mixing desk. Digital desks allow for all types of plugins to be used. This includes high pass/low pass filters, compressors of all types, speaker emulations, effects, etc. In this case, the sound person will usually want a basic signal.
All this allows for a lot of different options and ways to approach the tasks.