The terms "bull" and "bear" market really refer to the future of the market more so than the present. You will also often hear that we've been in a "bull" or "bear" market after the fact. So, right now, you could say that we're in a bear market, but that would normally mean that you expect the market to continue down.
A recession, however, will come with a bear market and many will call it a recession if the market falls about 20% from highs. We are about 10% from highs now. If you believe that this is a true recession, you are likely bearish about the market and feel that we are heading for another 10% decline. If you are bullish, you may call this a 10% "correction" and expect that we'll start back up from here.
So, those terms really have to do with the trend and the expected direction of the market more than the market's present standing. That's why, even after a 10% drop people will ask the question of whether one is bullish or bearish about the market.