A number of answers here would not have passed. Thankfully, none of the students said the difference was based on some conscious/unconscious fake dichotomy
Whether an experimenter recognizes and can verbalize his own bias (conscious bias) is certainly irrelevant to the outcome. I can't call this a false dichotomy so much as an irrelevant dichotomy. Personally I think JeffC has the main point; the OP's is a semantic question, so why create language constructs what can only lead to silly paradoxes, misunderstandings and to no other good ?
I think any definition of "confirmation bias" must include all selection bias regardless of some hypothetical psychological state. And hasn't the data been "cherry picked" even when the experimenter lacks self-recognition of bias ? It's a pointless and dubious distinction.
On a less pedantic, but related point, I once gave a talk to a group of beer judges (a serious hobby for me) in the issue of bias, trying hard to soften the blow with the phrase "unconscious bias". I was treated as tho' I had called them all a pack of liars & cheaters. One fellow in particular gave me a heated rebuttal on how the judges were above any bias, and how I was a jerk for suggesting otherwise.
Not only are biases real, common and often obvious, but there seems to be an extremely strong psychological feature that generally prevents people from recognizing or even considering their own biases.