Another partial answer based on Kurt Volker, who is the United States Special Representative for Ukraine Negotiations. While not serving under the president, he serves under the Secretary of State.
The Wikipedia article on United States Special Envoys characterizes these positions in general as:
Special Envoys, Representatives, Coordinators, and Advisors representing the federal government of the United States. These officials typically report directly to the United States Secretary of State. They normally require confirmation by the U.S. Senate. In some cases, when the incumbents have received earlier Senate confirmation to serve as an ambassador, that is not done. This can be particularly important when assigning an official to deal with a breaking crisis or conflict. In contrast to ambassadors who are responsible for the U.S.'s bilateral relations, many envoys oversee a portfolio that cover a broader goal or issue.
In his testimony before Congress on November 20th, 2019, he testified, according to a transcription by rev.com (emphasis mine):
Kurt Volker: (21:45)
My last three positions before leaving the senior foreign service in 2009 were as director for NATO in West European affairs at the National Security Council, principal deputy assistant secretary of state for European affairs at the state department, and finally, as US ambassador to NATO. In the spring of 2017 then secretary of state, Tillerson, asked if I would come back to government service as US Special Representative for Ukraine negotiations. I did this on a part-time, voluntary basis with no salary paid by the US taxpayer simply because I believed it was important to serve our country in this way. I believed I could steer US policy in the right direction. For over two years as US Special Representative for Ukraine negotiations, my singular focus was advancing the foreign policy and national security interests of the United States.