One way of answering this question is to switch the focus from the word "mean" to the word "meaning". Here is the definition of "meaning" in dictionary.com
1. what is intended to be, or actually is, expressed or indicated; signification; import:
the three meanings of a word.
2. the end, purpose, or significance of something:
What is the meaning of life? What is the meaning of this intrusion?
Referring back to the examples in the earlier What does it mean + to-infinitive / gerund
- To be an American / Being an American means having been born in the US.
It is a scientific type definition. Literally, X means Y, or X=Y.
And, the second example
- To be human / Being human means caring about others.
This is figurative. So, "entail", or "involve", or "imply".
For your sentences:
What does it mean to know the word?
What does it mean to use a method?
what does it mean to solve an equation?
It may just be my interpretation, but I view all of these from the scientific angle of X=Y. In these cases, I would interpret "what does it mean", to be the equivalent of "please provide a clear explanation, like a dictionary definition, for... how you go about solving an equation."
I don't think this is regarding importance, significance, purpose, implication, or anything figurative. No. It's simply a question of the very plain "meaning".
But that also depends on context. It could conceivably mean the other meaning of mean too.