How should LinkedIn summary look like for a person seeking PhD positions?
Look up some of the professors you are applying to. Unless they work in fields that have strong connections with industry, they will probably not have a profile at all; and if they don't have a profile, they won't even think of looking you up on it.

Even if they do, your application has a lot more information about you, and structured in a way that is more familiar to them (since everybody that applies has an application, but mostly do not have a linkedin account), so it is quicker to look up the information they may interested in.