Your definition is correct, and in reference to anaerobic bacteria, like
C. botulinum, that's what it means - no oxygen. Your smoker is never truly anaerobic, but heated, it does become a low-oxygen environment above the heat source. It isn't the smoker that has to be anaerobic for botulism to spread. The only anaerobic environment that matters is the micro-environment immediately surrounding the spores, which could be inside a sausage, or a canning jar, or inside a cut in a whole muscle meat, or under loose skin, between flesh and bone, along connective tissue between muscles, or in a remote spot inside a cavity, such as the body cavity of a fish being smoked or a puncture in muscle. The micro-environment that has to be anaerobic is so small that it isn't even observable, so it takes very little space without oxygen for the spores to proliferate if the pH, temperature, salt content and water activity are also within limits.
In the case of vegetative forms of
C. botulinum, they infect cells and have an anaerobic environment inside the cells, where they proliferate and produce toxin until the cell ruptures and the toxin is spread.
Here is a detailed discussion about
C. botulinum.
