Nomenclature of X-ray transitions-Siegbahn Notation

It is based on intensity: K$_alpha$ is the strongest line, K$_beta$ the second strongest, K$_gamma$ the third strongest. Yes, in this case it is from the next higher shell (from $2p$, $3p$, $4p$ etcetera).

For L$_alpha$ and L$_beta$, again L$_alpha$ is the stronger one, but the intensities are more similar than between the two strongest K lines. And both of these are transitions from $3d$ orbitals to $2p$. The difference is due to spin-orbit coupling between $2p_{1/2}$ and $2p_{3/2}$. And the much weaker third line L$_gamma$ is due to transitions to a $2s$ hole. Manne Siegbahn could not know this in 1916.

So there is no physics in the Greek subscripts of these two series (or of the M series). The nomenclature was purely phenomenological.

I notice that you want a source. This should be authoritative:

The nomenclature used for X-ray emission spectra was introduced by M. Siegbahn in the 1920's and is based upon the relative intensity of lines from different series. It gives no information about the origin of these lines. Since it was introduced, a number of lines have been observed which have not been classified within the Siegbahn nomenclature, particularly for the M and N series. Another problem is that its unsystematic nature makes the nomenclature difficult to learn.

R. Jenkins, R. Manne, R. Robin and C. Senemaud, Nomenclature system for X-ray spectroscopy, Pure & Appl. Chem., 63 (5), 735-746, 1991