Technically a C11 is as you indicated,C-E-G-Bb-D-F. Many eliminate the E to avoid dissonance, however that is an option and not a rule. Thus playing it either way is legal.

PS - it really depends on your ear and the melody. Sometimes having the E sounds good and sometimes it doesn't. YMMV

Whenever I get something stuck in the back of my throat, I dislodge it by drinking a beer.
It's called the Heineken Maneuver.

64 bit Win 10 Pro, the latest BiaB/RB, Roland Octa-Capture audio interface, a ton of software/hardware