FWIW, son had his transcript plus scores (AP/ACT) in sealed envelopes with his name on the envelope. This allows the coach to see at a quick glance if academics meet the school minimums. We didn't put any baseball stuff in the envelope on the theory that the coach can see performance and will not judge performance by awards, accolades, and stats created/awarded by others.
He (we) had a handle on which schools would be represented at the event and targeted schools which met his (our) desired profile. IN ADDITION, he (we) were open to any coach who wanted to chat with him (I never participated in these conversations and managed to be far enough away that he was on his own during the conversations.) I kept one of the college books in the car so I could get a quick handle on schools which we had not targeted but seemed interested in him - lots of schools popped up that I (we) hasn't considered (today, a smart phone will do the same).
"Demonstrated interest" is a term of art used by schools In which a potential student's contacts with the school are monitored - be it on campus tours, written requests for information, listing on standardized tests, etc. SOME schools (but by no means all schools) give some weight in the admission process to kids who have "demonstrated interest." For regular students, the common data set (CDS) lists the various elements of a kid's "resume" which are considered by the school.
"Demonstrated interest" - the term of art - isn't a factor for athletes (it's only used by admissions in weighing applicants); but a more layman's understanding is, I believe, what the coaches refer to. That simply means the kid has expressed interest to the coach that the school is on the kid's radar. For example, in a conversation (or even email) the fact that the kid's parents are Alma Maters, indicate the school is not a random selection. Basically, as used in this context, the coach is looking for players he has a legit shot to land if academics, baseball skills, and demand match up - as opposed to a random school.
Of course, if the coach sees the kid has an interest in the school, coach has a leg up; however, make no mistake, if the kid has something to show, coach will at least try to cultivate a relationship.
During S's recruiting (mostly as a rising senior), I would bet that half the schools which contacted him weren't even on the family radar; many would have been great matches. He had demonstrated no interest in any of them at the time of initial contact.
Probably the best way to show a coach a kid is truly investigating the school, is to show up on campus and speak with the coach (any coach, even a volunteer). If a kid is lucky enough, someone will be on campus. He met a dozen coaches using that approach - in their offices, for one on one sessions (parents were nervously walking the halls, out of sight, out of mind, waiting to debrief.) Even if no one was there, just talking his way into the building added to his ability to interview. (We began serious efforts to meet coaches as a rising sophomore; by then he had one year of grades which also showed a whiff of his course rigor.)