How to read Maxwell's Equations?

To think of one field "generating" or inducing another is not correct. The electromagnetic field is one entity. A changing magnetic field co-exists with a curling electric field. There is no time-delay, at a given point in space, between the existence of a changing B-field and the existence of an E-field with a non-zero curl.

In response to the further question. I didn't mention causality, which only enters the picture when changing currents and charge densities are introduced. Maxwell's equations can be used to derive wave equations of the form $$\nabla^2 {\bf E} = \mu_0 \epsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2},$$ $$\nabla^2 {\bf B} = \mu_0 \epsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2},$$ which show that the E- and B-fields both propagate at exactly the same (finite) speed $(\mu_0 \epsilon_0)^{-1/2}=c$.

In your question, there is the underlying issue of causality. Does the changing magnetic field cause a curl in the electric field or does the curl in electric field cause the magnetic field to change? It can be useful to paint a mental picture where one thinks of one thing causing the other, but in reality these two things are completely intertwined and can't be separated.

To understand the philosophical point better, please take a look at the writings of Ernst Mach, where he writes (analogously) about Newtonian mechanics and the concept of force as being an agent that causes acceleration as unnecessary.

https://en.wikipedia.org/wiki/Causality_(physics)

Regarding the edit, note that the answers say that E and B are intertwined at each point in space. How E and B at point X influence E and B at point Y as a function of time is another question entirely. Say you switch on a current in a wire in a space empty of electromagnetic fields. At the wire, you will get E and B fields, determined by Maxwell's equations, as you now have a current source (which is time-varying). These will create inhomogeneity in the E and B fields in space as there are fields at the source, but no field away from the source. This gives non-zero spatial and time derivatives of the components of the E and B field. You need to solve the field equations to understand at what speed and how this will propagate. The field one meter away from the wire does not instantaneously become non-zero as all spatial and time derivatives of the fields are initially zero and there are no sources at that location. You need to wait until there are variations in the fields just before the 1 meter distance, ie introduce time derivatives and spatial derivatives, to start "feeling" the influence of the source. This is where the finite speed of propagation of disturbances in the field comes from.