I'm trying to better understand Harmonious Academic Methodology (a Lore charm).
It can be used to train others, but the rules states that the lawgivers can also train himself.
So can a character with HAM train himself up to 4 dot in the allowed abilities spending 1 week of training for dot without using XP?
Not without using XP
From the same wiki, description of the Training Charm-keyword :
though they may award competence quickly and easily, the character must still pay for what they have gained.
From the HAM page you linked: "She can train them"... "or up to her own own ratings, whichever is lower" (emphasis mine). Any point she might try to gain would always be raising her rating to a level she doesn't already have it up to, and therefore would always be beyond the ability of HAM.
All effects in Exalted with the Training keyword require that the recipient(s) expend the appropriate amount of XP. If the target doesn't have enough XP, they go into XP debt. When a character with XP debt gains XP, half of the gained XP is spent buying off the debt. A character with XP debt cannot be the target of a training effect.
Thus, while you can use training effects to get around some of the more laborious training times, you can't use them to get around the associated XP costs.
Per 2e core rules, pg 183: (and I can't find a mention of any change in the errata 2.5)
Keyword "Training: This Charm trains other characters. Characters who receive training and aren’t extras must pay the normal experience cost for any trait increases. If they can’t afford this cost, they go into “experience point debt.” They must pay at least half of all awarded experience points, rounded down, toward this debt until repaid."
There is no mention of not being able to be the target of further training effects while in xp debt.
Harmonious Academic Method & Tiger Warrior both implicitly state: "... This Charm increases that trait for each member of the unit by one dot, to a maximum of 4. The Solar can train with the unit or as a solo unit, increasing her own traits. She cannot increase others’ traits past her own." << This indicate she can train her own traits higher as a solo unit up to a maximum of 4 (there is nothing is the 2.5 errata concerning this Charm). Then she can train others to a 4 (for either Charm).
The best answer though is do what makes you and your group happy. If players go this method, they may be looking for bigger challenges, or they may feel underpowered, if you're the ST talk to your group regarding why they'd want to go that route.
I've had personal experience with a group using those Charms (in that way) and the ones that later modify them to do more, it's just playing at a different level, the limits of Age (and then Essence) still apply. There's also the question of how much down time the group has to actually get even 5 hour of training time in a week, that's up the ST really (unless they want to ignore the Abyssal that's trying to kill them, because study group is more important).
