I've always understood that there are three types of different types of university - private, public and corporate.
Private institutions are exactly that - educational establishments that can award degrees in certain subjects. The best examples of these are the theological colleges, some church sponsored, some sponsored by individuals.
Public universities are the ones that most people think of - Oxford, Cambridge, John Hopkins, etc. Award a wide range of degrees, but many have specialisations in key subjects (e.g. languages, engineering, medecine, literature, etc), attracting students from any background who wish to excel within those fields and subfields.
Corporate universities are exactly the same as public universities, offering similar courses, but usually only open to the children of employees in good standing and possibly offering education to promising outsiders for a "return of service" (i.e. earn a bachelor's degree in biotechnology from the "University of Ares" for a ten year employment contract).
There are certain cross overs as well - it's well known that certain corporations and organisations sponsor entire departments of public universities. The corp gets to review the most promising students whilst the university gains extra funding. Win-win.
As for extractions, certainly - especially if a student has alrady signed a "golden handshake" with another organisation. Same with professors and researchers. They'll also try other dirty tricks - counter offers (better contract & facilities), blackmail, etc, etc.
