Relationship between three different definitions of solutions for ODE with irregular coefficient

What is the difference between the notions of

of an ODE $$\dot \Phi(t,x) = b(t,\Phi(t,x))$$, with initial condition $$\Phi(0,x) = x$$, where $$b$$ is a non-smooth (that is, non-Lipschitz) vector field?

In general, are they related in any way? Under which (non-trivial) assumptions are they equivalent? Which is the strongest notion of solutions between these?