I just want to chime in to remind folks that what sc_member and SearchCommitteeChair said apply to research universities. A good cover letter for a slac or any other place that emphasizes teaching (e.g. most regional universities) will probably be longer and will be structured differently. Two pages single-spaced is still a good limit to aim for, but depending on your formatting you could go over, especially if you have a lot of experience so there is more to say.
Intro / interest in the position - Para 1 (background is optional here IMO because that's going to be on the first half of the first page of you c.v.; only as it pertains to introducing yourself and why you fit the position)
Teaching interests / experience, using the school's course titles - Para 2
Teaching philosophy - Para 3 (2 and 3 can be one para if the interests and experience are short)
What your research is and how it's suitable to involving undergraduates, as well as what kinds of mentorship you have done for undergraduates - Para 4-5 or 4-6
Other stuff that is special about you for this job, including aspects of fit, service experience, diversity/inclusion experience, etc - Para 7
What is in the application and who the letters are from / conclusion - Para 8
So I'd say it's not just about moving the teaching stuff first, it's also how much space you devote to it, and perhaps more importantly, how you talk about your research in a student-focused way. You might also find that you need an extra paragraph for one of these topics, depending on your experiences.
Generally speaking service shouldn't be mentioned in job applications except as c.v. lines but in a practical sense, if the place you're applying is not very elite/wealthy, the faculty will be doing lots of service. And they won't want to be stuck with a colleague who doesn't pull his/her load. This is because less wealthy places have higher student-faculty ratios, so there are fewer faculty for the service jobs to go around to. Faculty also sometimes end up doing tasks under the umbrella of "service" that are completed by paid staff members elsewhere. So being a team player and someone who can do service is super important at places like this. It should never be an emphasis of the application, but if you're someone who has done more than your fair share and/or who has enjoyed getting particularly involved, then you should consider finding a way to say it at the end of your cover letter.