EXEC1F is the c/s assigned to whatever plane the first family, the spouse or children of the President is flying on.
The aircraft used the EXEC1F callsign when it dropped off the first family in Delaware.
When it left Delaware it used SAM### callsign indicating it was carrying other VIPs, senior military or government officials.
EXEC1F is never used by the president. EXEC1 is only ever used if on the very rare occasion that the president flys private civil aircraft.
You are mixing up two completely different though related callsigns.
The plane 01-0015 is a C-40 military VIP transport plane.
When the plane was using EXEC1F it was operating under military status as clearly shown under DB flags.
Your sources are poor and you need better sources.
The cap is from official FAA document regarding use of EXEC and other callsigns.
https://www.faa.gov/air_traffic/publications/atpubs/atc_html/chap2_section_3.html