It does, you are right. It can refer to the number of terms in the academic year. A semester divides the academic year into two terms, a trimestre into three. In this sense, a trimestre can be as little as eight weeks. And a quadmester, or a 'quarter', divides the academic year into four terms. It all depends on the context with which you use these terms (business vs. academic vs. medical).
A trimester of 8 weeks is actually a half-trimester.
Confusing for sure, in part because logically it should be called a setrimester. =]
But in the global south, they break down semesters to two terms each!
If the particular trimester being referred to is regarding a [human] pregnancy then it does not matter when the trimester started. Pregnancies do not begin at set times of the year.
But the reality is, that a trimester (or quarters if you prefer) for any reason can start at any time (schools in the northern hemisphere tend to start in August, September, and even as late as October), but the basic rule that applies to trimesters in general is the 3 month thing, if talking about out of a year.
Many universities have a fiscal year which ends during the summer, both to align the fiscal year with the academic year (and, in some cases involving public universities, with the state government's fiscal year), and because the school is normally less busy during the summer months. In the northern hemisphere this is July to the next June.