Motivation for considering the upper numbering of ramification groups Let $L/K$ be a finite Galois extension. We denote by $G_s$ its $s$-th ramification group. Define the Herbrand function $$\eta_{L/K}:[-1,\infty) \to [-1, \infty), \ \eta_{L/K}(s) = \int_0^s \frac{1}{(G_0:G_x)} \ dx.$$ Let $\psi_{L/K} : [-1, \infty) \to [-1, \infty)$ be its inverse. Then, introducing the upper numbering $G^t = G_{\psi(t)}$ has many uses.
This is great but what is the motivation to actually consider this numbering? I'll admit that my intuition on ramification groups is still quite weak but for me it seems a bit out of the blue. Is it just because Herbrand's Theorem $G_s(L/K)H/H = G_{\eta_{L/L'}(s)}(L'/K)$ for an intermediate field $L'$ appears more natural, so one would try considering a different numbering? Is there a different motivation?

The short answer is that lower ramification groups behave well when taking subgroups, while upper ramification groups behave well when taking quotients. As a result, the upper numbering can be defined for infinite extensions. I think this is the key motivation for defining them.
Indeed, if $L/K$ is an infinite extension, we can define
$$\mathrm{Gal}(L/K)^u = \varprojlim_{M/K\ \mathrm{finite}, M\subset L}\mathrm{Gal}(M/K)^u.$$
This definition makes sense by Herbrand's theorem, which tells us that we get a projective system to take the limit of.
In this way, the upper ramification groups are far more natural than the lower ramification groups.
The lower ramification groups have the advantage that they are easier to define and are sufficient for finite extensions. They also behave well with respect to subgroups: if $L/M/K$ is a sequence of Galois extensions, then
$$\mathrm{Gal}(L/M)_i = \mathrm{Gal}(L/M)\cap \mathrm{Gal}(L/K)_i.$$
