> How can one meaningfully say that one field generates the other in an EM-wave?

You can't because they don't. The electromagnetic wave is an _electromagnetic_ field variation. See [Wikipedia](https://en.wikipedia.org/wiki/Electromagnetic_field#Dynamics) where you can read this: _"Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole — the electromagnetic field"._ 

Or see section 11.10 of Jackson's [Classical Electrodynamics](https://archive.org/stream/ClassicalElectrodynamics/Jackson-ClassicalElectrodynamics#page/n397/mode/2up) where he says this: _"one should properly speak of the electromagnetic field F<sub>μν</sub> rather than E or B separately"_. 

Or see Jefimenko [here](https://en.wikipedia.org/wiki/Jefimenko%27s_equations#Discussion): _"...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents"._ 

It's a popscience myth that an E wave generates a B wave which generates an E wave. One of those ["lies to children"](https://en.wikipedia.org/wiki/Lie-to-children) that ends up being widely believed, even by professional physicists.   

> Clearly there are nuances of how one states the "mutual induction" explanation for EM-waves. 

I'm afraid there's no nuance at all Julia. The "mutual induction" explanation  is wrong. E and B are in phase because [they're space and time derivatives](https://en.wikipedia.org/wiki/Electromagnetic_radiation#Derivation_from_electromagnetic_theory). See [this answer](http://physics.stackexchange.com/a/223989/76162) where I used a canoe analogy.   

> My question is, of how strong can the statement in this direction be if one insists on that there should be a mathematical proof for it starting from Maxwell's equations. Nevertheless the statement should be simple enough to serve as (correct  but) "popularized" view on the EM-field and it should be close to what is often said about EM-waves and mutual generation of the field in undergraduate and high school physics courses.

There is no mathematical proof. The equals sign in $\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} $ and in $\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\ $ does not indicate cause. Instead you should read it as "is another aspect of". Or simply "is".  

> If one says that one field generates the other, one should distinguish between a causal relation and a logical one (assuming Maxwell's theory). 

Maxwell unified electricity and magnetism to give us the electromagnetic field. But here we are 150 years later and people still will talk about E and B as if they're two totally different things. Tut.      

> Let's start with the second (I guess weaker) interpretation: From Maxwell's equations in vacuum: $\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} $ [and] $\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\ $ one can see that if one field is changing with time, logically necessary (assuming the validity of Maxwell's equations) there must also be the other field (not necessary both in the same point, but somewhere in space, i.e. it can not be identically zero since this would imply that the curl is zero and thus also the time derivative of the other field). 

There aren't really two different fields, there's one field, and two derivatives of it. Spatial and time. 

> This seems to be a very weak interpretation of the mutual generation thing.

Agreed. When you look into it it just doesn't stand up. 

> One guess to make the statement stronger may be not to talk about existence but say something like: If one field $\mathbf{E}$ is changing with time, there must be a $\mathbf{B}$-field which is also changing with time (and vice versa). 

That's not quite right because there's only one field there.  

This seems to be wrong...

It is.  

> Up to now this is all logically not causally. How can one interpret the statement in a causal way such that it is correct? 

You can't, because it isn't. 

> Since electric and magnetic fields are "just" different components (in a fixed reference system) of the electromagnetic field tensor I guess that there will be no correct causal statement at all, but I am not sure.

There is no correct causal statement at all.