1. preamp/amp modelling, DI, FOH
2. your own rig, mic, FOH
if you are using borrowed equipment it would be too much to ask to get the same kind of tone everytime. amps and cabs colour the tone a GREAT deal.
if the venue is small AND with very good acoustics, i would bring my own amp and cab. my cab sucks, but i at least i know how to work a tone from it.
if i were to run a cable between the balanced output of my amp into the soundboard, i would use the monitors for monitoring. so i wouldn't even use a cab. this is to ensure that the sound that comes to me is as accurate a representation of the sound that goes to the audience as possible.
if i were to use a mic'ed cab setup, i would still use the monitors. additionally, i would prefer to block the sound that is coming from the cab. another benefit of this is to avoid interfering with other instruments that are mic'ed. but this is a rather complicated setup. i never go to such lengths to set up in such a manner, and i would never expect anyone to patronise such requests either.
if you want serious tone be prepared to get serious equipment and spend serious money. anything else would be classified as making the best of the situation. if not, beggars can't be choosy