"No meetings != no communication" is analogous to saying "Without meetings, people may still be communicating."

"Meetings != communication" is analogous to saying "At meetings, communication doesn't necessarily occur."

In terms you would personally appreciate: (No Patrick Hernandez != No Dancing) similarly is not equivalent to (Patrick Hernandez != Dancing)

One says without Patrick Hernandez, there may be dancing. The other says with Patrick Hernandez there certainly is not any dancing ;-)