Different notions of equivalence of norms

Question

I would like to understand deeper the notion of equivalence of norms, that can be understood in the sense that there are constants such that $$(1) \qquad aN_1 \leqslant N_2 \leqslant bN_1$$

or that there is an exponent such that $$(2) \qquad N_1 = N_2^\alpha$$

What are the motivations for these choices depending on the interests? In particulier, is it clear that the Cauchy sequences are preserved if and only if the relation $(2)$ holds?

Answer 1

The question has been answered by Michael Greinecker's comment.

I wrote this answer to make it visible at first glance that the question is no longer open.