I think this is a matter of preference, comfort and the situation. My just son didn't feel comfortable talking about other schools to other coaches. He's not a talker to begin with, and this just put him in an awkward spot. So, when the topic of other schools came up he would mention whether it was D1 or D3 and their conference....that was it. He kept it very high level, and he would share his interest level in that school at that moment. What the schools/coaches didn't know was he was basing his interest level in their engineering program. He kept that to himself.
So, I don't think there is a right or wrong answer here. Just what you are comfortable with.
As always, JMO.