How to minimize a FSM transducer? In contrast to FSM minimization which is well studied with various algorithms, theorems and has many practical applications, I'm looking for any nontrivial insight, analysis and references to the following question.


  Given a FSM transducer, how does one find a "minimal" equivalent transducer?


Note that there are possibly multiple reasonable ways to define the minimal transducer. The literature maybe hasn't settled on one. Also, I'm looking for applications if some are known.
As you pointed out, there are several ways to define minimal transducers, but I only know of two mathematically appealing definitions.

The first result concerns the reduction of linear representations of recognizable series
(= defined by weighted automata). The best reference is Chapter II, Minimization, in one of these two books (the more recent is an update of the first one, with many new additions):

[1] J. Berstel and C. Reutenauer, Noncommutative rational series with applications, Encyclopedia of Mathematics and its Applications, 137. Cambridge University Press, Cambridge, 2011. xiv+248 pp. ISBN: 978-0-521-19022-0

(2] J. Berstel and C. Reutenauer, Rational series and their languages, EATCS Monographs on Theoretical Computer Science, 12. Springer-Verlag, Berlin, 1988. viii+151 pp. ISBN: 3-540-18626-3

In the case of sequential functions, one can adapt Nerode's construction to obtain a minimal sequential transducer. The algorithmm was first given in

[3] G. N. Raney, Sequential functions. J. Assoc. Comput. Mach. 5 (1958) 177-180.

See also Thm XII.4.2 in

[4] S. Eilenberg, Automata, Languages, and Machines. Vol. A, Pure and Applied Mathematics, 58, Academic Press (1974)

The algorithm was extended to so-called subsequential functions by Choffrut. See the survey

[5] C. Choffrut, Minimizing subsequential transducers: a survey. Selected papers in honor of Jean Berstel. Theoret. Comput. Sci. 292 (2003), 131-143. 

You may also have a look at
A tutorial on sequential functions for an introduction.