I was told the framing of a knot is the linking number of the push-off. But I don't understand why the framing does not depend on the knot but only on the parallel copy.
How about a Legendrian knot? (I heard this is more rigid than a topological knot)
A framed knot can be imagined as a knot tied in a ribbon with width, rather than a 1-dimensional string. We also want the ribbon to link back up to itself exactly, so you can imagine one edge of the ribbon is colored blue and the other is red (no Möbius strips allowed).
The self-linking number of the framed link is the linking number of the red curve with the blue curve. It's not "just the parallel copy", it's a relationship between the parallel copy and the original.
If you just have a knot diagram sitting in the plane, you can look at the blackboard framing: what if you just let the ribbon lay flat in the plane. The linking number of a blackboard-framed link can be computed as the writhe (sum of crossing signs) of the diagram.
I believe two framed knots are the same as framed knots if and only if they are (1) the same as knots (forgetting the twisting of the ribbon) and (2) have the same self-linking number (number of twists of the ribbon).
Framed Knots seems like a pretty decent reference.
I don't know if/how Legendrian knots are related.