After some numerical experiments I indeed come to the conclusion that

matrix power tetration and intuitive tetration are equal.

For base <= eta both are equal to regular tetration.

For base > eta both are equal to Cauchy tetration.

If they are equal to regular tetration how do they know how to choose the right (lower) fixed point?

This depends on the location of the development point of the method.

If the development point lies inside the basin of attraction of the fixed point, then it converges to the regular iteration of that fixed point.

E.g. the lower fixed point on the real axis is the only attracting fixed point of exp_b at all.

For b = sqrt(2) you can for example choose the development point to be the imaginary unit and despite the values on the real axis have vanishing imaginary part.

This would imply that the by some of the above methods iterated function does not continuously depend on the development point but it stays the same function for all points of the basin of attraction and suddenly jumps chaotically/is non-convergent as long as it is in the Julia set of the function.

This behavior is very similar to the behavior of the limit formulas or the Newton&Lagrange formulas for regular iteration. To which fixed point the regular iteration belongs is just determined by in which basin of attraction the initial value x0 lies.

Particularly in the parabolic case where there are petals of attraction and repulsion at the *same* fixed point, which yield different regular iterations. For exp_eta there are two petals the lower one is attracting, the upper repelling.

Somehow one can imagine infinity perhaps also as such a fixed point, which comes into play for b > eta. The basin of attraction is the whole complex plane, so the development point should not be important.

However there are still some unresolved questions. If we - for b=sqrt(2) - develop at some point above the upper fixed point, do we obtain then the regular iteration at the upper (repelling) fixed point? And if so, how does the method know to choose this repelling fixed point though there are so many more repelling fixed points in the complex plane? Why does a non-real development point not work (according to Jay) for b>eta?

matrix power tetration and intuitive tetration are equal.

For base <= eta both are equal to regular tetration.

For base > eta both are equal to Cauchy tetration.

If they are equal to regular tetration how do they know how to choose the right (lower) fixed point?

This depends on the location of the development point of the method.

If the development point lies inside the basin of attraction of the fixed point, then it converges to the regular iteration of that fixed point.

E.g. the lower fixed point on the real axis is the only attracting fixed point of exp_b at all.

For b = sqrt(2) you can for example choose the development point to be the imaginary unit and despite the values on the real axis have vanishing imaginary part.

This would imply that the by some of the above methods iterated function does not continuously depend on the development point but it stays the same function for all points of the basin of attraction and suddenly jumps chaotically/is non-convergent as long as it is in the Julia set of the function.

This behavior is very similar to the behavior of the limit formulas or the Newton&Lagrange formulas for regular iteration. To which fixed point the regular iteration belongs is just determined by in which basin of attraction the initial value x0 lies.

Particularly in the parabolic case where there are petals of attraction and repulsion at the *same* fixed point, which yield different regular iterations. For exp_eta there are two petals the lower one is attracting, the upper repelling.

Somehow one can imagine infinity perhaps also as such a fixed point, which comes into play for b > eta. The basin of attraction is the whole complex plane, so the development point should not be important.

However there are still some unresolved questions. If we - for b=sqrt(2) - develop at some point above the upper fixed point, do we obtain then the regular iteration at the upper (repelling) fixed point? And if so, how does the method know to choose this repelling fixed point though there are so many more repelling fixed points in the complex plane? Why does a non-real development point not work (according to Jay) for b>eta?