To back you up, I can confirm that people who spend their time thinking a lot about this stuff (political scientists and those in related fields) spend zero time quibbling over the use of the term "democracy" as it's commonly used, and are likely to eye-roll and attempt to disengage from conversation should someone push their glasses up their nose and "well, actually..." the issue. In most contexts they use democracy as an umbrella term, the same way everyone else does. Doing otherwise marks one as an outsider to the field, or a novice—any in teaching positions get to correct plenty of "clever" freshmen on the issue every year.