A vested right shall be deemed established with respect to any property upon the approval of a site-specific vesting plan, pursuant to the provisions of this Section. Such vested right shall confer upon the landowner the right to undertake and complete the development and use of said property under the terms and conditions of the site-specific vesting plan. This right to carry out the development in accordance with the approved plan shall attach to and run with the land, rather than being personal to the recipient of plan approval.