Deep engagement in a conversation and a deep conversation are different things I'd say.

Deep engagement to me is when someone starts thinking about your position as their own. One time I asked a store clerk where I can get a shovel in the store. They didn't have any, but he kept brainstorming with me what I can use as a makeshift shovel or where else I can go to shop for one. It was very engaging and nice to be part of.

One way I can describe a deep conversation is a topic that, when someone starts getting into it, the socialized knee-jerk reaction is to insult them to shut them up (unless you happen to be impassioned about it as well). Think sitcom: some quirky character waxes poetic and the others tell them to can it because the plot must go on.

I guess a deep conversation can be a personal one, although I would maybe categorize that as an intimate conversation rather than deep. Both are conversations that people usually just to ignore, avoid, or tell others to stop because they want to get on with their own lives. Usually deep conversation topics are larger-than-daily-life topics, so that's probably why