Emitting vapor is included in definitions of the word smoking, so technically, vaping can be considered smoking. I don't see it that way, though.
What is the name of this section of the forum?
And I don't agree with the section name either. For that matter, the name of the entire website.
(Please tell the mods to stop looking at me that way. They're scaring me.)
Anyway, that's sideways to my point. Context is everything. "No smoking" is referring quite clearly to the combustion of tobacco. The "emit visible vapor" to which you refer has to do with what is called the "smoke point" of oils in cooking. That's not really combustion. If oil actually combusts in your kitchen, you need to grab a fire extinguisher or run. Actual smoke form burning oil is something else all together and Very Much Not Good.
Notice, for instance, nobody says, "the tea kettle is smoking". Even though it admits visible vapor. You don't read recipes that say, "boil the water until it smokes" even though it emits visible vapor. Vaping is much more akin to boiling water than burning tobacco.
(By the way, splitting hairs is a hobby of mine. Also, had Katrina never happened, I would probably be working on my Ph.D. in anthropological linguistics about now. Before the disaster, I was bailing on computers and trying to move to my true love: language. Then... well... stuff happened. Anyway, fair warning, I can talk language so long, there is a risk of coma to the listener. There has been talk of tattooing a warning label on me.)
Point being, that connotation of "smoking" doesn't even begin to approach vaping in any way, shape or form. And "no smoking" signs are clearly aimed at the burning of tobacco. As in the case of the OP, the restaurant found "chewing" to be acceptable. That's using a tobacco product but not burning it. They, themselves, did not see the "no smoking" sign applying to that use of tobacco.
Applying "smoking" terms of just about any kind to vaping is an artifact of it's creation for and adoption by smokers. We just carry the existing terminology over instead inventing a whole new vocabulary (though that is in progress also, just ask any newbie over in "New Members" how confusing we are at first

).
The actual laws behind smoking bans do not, yet in most all cases, address vaping in the least. If you're rich and want to fight, you could probably beat a citation for smoking in a "no smoking" area. In this context, "smoking" has a fairly clear meaning involving burning tobacco.
Now, a private business can--within the confines of law--create and enforce pretty much any rules they want. It may, however, not be bright on their part on the economics side of things however. A restaurant that allowed chewing and spitting? I ain't goin'. I've always found that to be a pretty vile habit. It's a bit bone headed on their part to allow that over vaping.
Though the food service business isn't exactly filled with legal scholars. They see smoke looking stuff come out of a person, they think "smoking". And with the prevailing legal atmosphere of smoking being banned so much, I'm not surprised some would overreact. Or call it "err on the side of caution" if you feel generous.
(Still, ultimately, the OP had every right to "vote" with his money and go elsewhere. That is supposedly our system of "capitalism".)