Isn't the bifurcation of a system, say $\dot{x} = \mu \sin(x) - 2x$, related to the system rather than a fixed point, say $x^{\star} = 0$?.
My understanding is that a bifurcation occurs when there's a "qualitative change" in the derivative vector field in the graph $(x, y) = (\mu, x^\star)$, where $x^\star$ is the solution to $\dot{x} = 0$ for a particular $\mu$. But this understanding of a bifurcation point depends on the system, not a particular fixed point of the system.
Is the bifurcation of a fixed point always at $x^{\star} = 0$? Is this by convention? (i.e. $x^{\star} = 0$ is always the fixed point of a system in normal form).
My current understanding is that (i) it only makes sense to talk of the bifurcation of a particular fixed point if the fixed point remains a fixed point for all values of $\mu$ e.g. $\dot{x} = x(x - \mu)$, (ii) that the bifurcation of a fixed point always occurs at $x^\star = 0$ for normal form systems, and (iii) that the bifurcation of $x^\star = 0$ can practically be understood as the point at which $x^\star$ undergoes a change in stability.
If I understand your elaborations correctly, they are correct:
A bifurcation is a point in parameter space where the stability of fixed points¹ changes or new fixed points¹ emerge. Calling this bifurcation of a fixed point is just a terminological shorthand that serves to describe the kind of bifurcation (contrasting it, e.g., to a bifurcation of a limit cycle) or specifying which fixed point participates in a bifurcation (in a system with many fixed points). Insisting that bifurcation of a fixed point does not make sense because the fixed point may not be one after the bifurcation is – to be blunt – semantic pedantry that ignores the real use of language.
Fixed points undergoing a bifurcation can be located anywhere in state space. That they are (usually) at the origin for canonical forms is just because canonical forms try to simplify what is happening as much as possible and this (usually) means moving the fixed point to the origin.
¹ or limit cycle, chaotic invariant set, …