[Answer to questions in the question body.]
Suppose I work at the White House.
Then you would have been notified that, beginning in January 2018, personal cell phones and other electronic devices are prohibited from the West Wing, except for the president.
The president tells me that he wants to make a phone call and not leave any record of it.
The president would use their own personal cell phone. If the call is personal, there is no requirement for record. If the call is official and no record is made, then it would violate the Presidential Records Act.
In July 2018, Business Insider reported that President Trump gave his personal cellphone number to various world leaders, having unrecorded conversations with them completely without U.S. officials' knowledge.
Is there actually a way to do this, or is there some system in place in which everything that comes in and out of the White House is logged by an independent body?
The president using a personal cell phone is not logged.
Are all presidential telephone calls logged at the White House?:
[The answerer notes having Worked in the White House for six years]
Yes and No. All Presidential calls that are made through the switchboard are logged both in and out.
But what he makes off his private line are not logged by the switchboard operators. Although we all know that records do exist of all calls and texts. His private calls would be very hard to gain access to. There is nothing to stop him from having a burner prepaid cell phone other than guidelines. ...
Is trying to re-classify calls as top secret the best that the White House can do?
The call that was re-classified was an official call and there was a record.