As to how the spirit fights, as GM I tend to be directed a bit by the details of the instruction, how popular the mage is with spirits in general, how it has treated that spirit in particular, the role that spirit type plays in the tradition in question, and even description of the spirit.
I don't track the spirit reputation (or whatever it is called) from Street Grimoire per se, but I keep a rough tally in my head of how the mage has been with their spirits. Mages with a reputation for taking good care of spirits tend to get more willing cooperation, where the spirit will use its intelligence to choose effective methods and smart strategy (I've never had a player mage be a bastard to spirits, but I've had NPC that I'd decided are, and their spirits tend to choose less effective methods, given a choice). And if the spirit has already been summoned for a while, how has the summoner treated it? Did they make a fire spirit materialize in the rain or an air spirit in a narrow tunnel? Did they greet it with some reagents to consume, or take the time to explain why its help was needed and praise its prowess?
There are variants on "fight those people" which can change things. One key possible modifier is "until you are seriously hurt, at which time you are free to depart." Knowing that they are not going to be made to fight until disrupted may make a spirit more willing to risk taking harm. But there are alternatives to "fight" and "those people". "Leave nobody conscious in this room but me" gives the spirit freedom to drive people out through fear, knock them unconscious, or kill them, for example. Or "fight any awakened who are currently between five and twenty-five meters of me." "Do whatever you can to keep any other spirits from attacking me" (best used with a spirit that is in a good mood so that they don't decide that the easiest way is to knock you out themself, or use fear on you to drive you out of line of sight, or something).
Not every earth spirit is the same. One factor in that is what role in plays in the summoners tradition. If earth is seen as a healing element, an earth spirit may be more reluctant to cause real harm, may fight more defensively, prefer things like confusion or fear, use stunning attacks, and so on, while if it is the traditions combat aligned spirit it may be far happier being an effective warrior.
And finally, if the player takes the time to describe the spirit, I'll take that into account. If the water spirit looks like a giant squid it seems like it might engulf or bind, an air spirit that looks like a thunderbird may prefer elemental attack, and so on.