Universities asking for proof of work experience want dates, designation and responsibilities that match what you claimed in your application. Ask for the responsibilities section to be detailed enough to support your admission essay.
An experience letter confirms what you did and for how long. A relieving letter confirms that you completed your notice period and were formally released. New employers in India and the Gulf frequently ask for both.
Practice varies by jurisdiction. Many employers issue one as a matter of policy, and some labour laws require a service certificate on request. If they refuse, offer letters, payslips and tax records are usually accepted as alternative proof.
HR or the people operations team, with your reporting manager copied. Managers rarely have the authority to issue letterhead documents themselves.