1) Speaking as a former truss designer, I favor phamENG's approach.
2) Yeah, the truss designer will eventually figure all of this stuff out for herself if she has to but she'll likely waste a lot of time doing it. And perhaps a lot of the design team's time too. Within the truss industry the delineation of responsibility is thought to be:
a) The design team defines the roof and ceiling planes into which the trusses fit. This nicely mirrors the fancy layout software that the truss folks use that has them model the "trussed volume" in 3D before the individual trusses go in.
b) The truss designer defines the profiles of the individual trusses based on their final locations and -- obviously -- the webbing configuration if the design team has to no weird preferences in that regard.
This makes sense based on who controls what in the process.
3) In my experience, most architects are not capable of fully specifying all of the geometric information that a truss designer needs. They're good for OH, pitch, and heel height if they're on the ball but that's about it. They usually aren't capable of thinking through the framing arrangement in the detailed manner that a good structural engineer can/will.
4) Generating the kind of profiles that phamENG describes takes very little time. And, in my opinion, it should probably be a nearly automatic outgrowth of the engineer thinking through the details of the roof system in the first place. In my experience, too many engineers treat pitched roofs as no different than flat roof systems when they lay out their framing. This can lead to all manner of poor layout choices.
5) In more complex, pitched roofs, there can be a number of possible layouts. If the engineer doesn't care what the outcome is then, so be it, the truss designer can do whatever suits them. If the engineer doesn't care what the layout is, however, then it strikes me as only fair that the engineer should provide enough direction to the truss designer such that the truss designer isn't burdened with a bunch of rework trying to figure out what the EOR wants.
6) To an extent, doing the truss profiles is simply a way to encourage coordination and decision making between the architect and the EOR. In many instances, overhang length and heel heights have structural implications for things like uplift and the best scheme for transferring diaphragm shear down through the heel height and into the shear walls.