(Ref AC 61-65H solo endorsements 61.87(n) and 61.87(p)). When should I give 61.87(n) vs 61.87(p)?
Remember the ACs are guides, not laws, as opposed to the FARs.
14 CFR §61.87(n) reads:

Limitations on student pilots operating an aircraft in solo flight. A student pilot may not operate an aircraft in solo flight unless that student pilot has received an endorsement in the student’s logbook for the specific make and model aircraft to be flown by an authorized instructor who gave the training within the 90 days preceding the date of the flight.

14 CFR §61.87(p) reads:

Limitations on flight instructors authorizing solo flight. No instructor may authorize a student pilot to perform a solo flight unless that instructor has—
(1) Given that student pilot training in the make and model of aircraft or a similar make and model of aircraft in which the solo flight is to be flown;
(2) Determined the student pilot is proficient in the maneuvers and procedures prescribed in this section;
(3) Determined the student pilot is proficient in the make and model of aircraft to be flown; and
(4) Endorsed the student pilot’s logbook for the specific make and model aircraft to be flown, and that endorsement remains current for solo flight privileges, provided an authorized instructor updates the student’s logbook every 90 days thereafter.

The upshot of which is that (n) is, technically the only requirement for the student to possess for initial authorization for solo flight by a student pilot. (p) is also applicable to the endorsement that must be met by the instructor for initial endorsement, but must be met for additional 90 day solo endorsements by the student.
Here’s the thing, as there is no legal requirement for exactly HOW TO PHRASE an endorsement; they can technically be written by a CFI, provided said endorsement conforms with §61.87, you’re perfectly welcome, as a flight instructor, to give an initial endorsement referencing both (n) and (p) if you desire.