I think there is a lot of talking past each other going on.
On one hand, no one is disputing the fact that the market has had some steep drawdowns since 2009. The bull market has not been a straight line upward. And, yes, literal definitions allow someone to say there was a bear market, or a correction. As a result, one might then say something like "how can we still be in the same market cycle when we experienced these correction/bear periods?"
On the other hand, folks talking about a continued cycle (as I have) are referring to index stock market valuations. While there have been drawdowns in price, the valuations have remained materially higher than historical median valuations (and I'm talking about more than CAPE; this is a pretty universal understanding across various valuation methods).
Seems like the two camps in here are kind of debating different things. Just my observation, for what that's worth