Which definition at http://www.oxfordlearnersdictionaries.com/definition/english/vest_2 applies to http://www.merriam-webster.com/dictionary/vested%20interest? I want to dredge below this term, and ask not about the Merriam Webster dictionary that I already apprehend.
The phrase vested interest has a different meaning in common usage than the individual words. Definition of vested interest noun from the Oxford Advanced Learner's Dictionary: "a personal reason for wanting something to happen, especially because you get some advantage from it". For example, "She has a vested interest in the proposed ban on plastic bags because she owns a business that sells canvas shopping bags."
The meaning probably evolved from the legal definition "A right or title, as to present or future possession of an estate, that can be conveyed to another." If you held a vested interest in a property, you would have a personal interest in increasing or maintaining the value of the property, so that your title to it is valuable.