...spin¹

Historically, the term spin had (and sometimes has) a narrower meaning than angular momentum. Picture an electron in an atom. Sacrificing some accuracy on the altar of visual imagery, think of the electron as orbiting about the nucleus, and spinning on its axis. The orbital motion gives rise to so-called orbital angular momentum; the spinning gives rise to so-called intrinsic or spin angular momentum.

... momentum²

And $l$ traditionally stands for orbital angular momentum, and $s$ stands for spin angular momentum.

... component.³

People sometimes use $m_z$ instead of $j_z$ to denote this.

... somehow⁴

Not so trivial, because the three component measurements are mutually incompatible; it is theoretically impossible to measure $j_x$ and $j_y$ simultaneously (for example). Nonetheless, the combination $j_x^2+j_y^2+j_z^2$ is measurable.

... corrections⁵

Specifically, Bohr's energy levels agree exactly with those derived from Schrödinger's equation, when we neglect the spin of the electron, the spin of the proton, and relativistic effects. (Spin itself is sometimes considered a relativistic effect.)

... above).⁶

Sommerfeld gave a single more general quantum condition which implied the quantum conditions for $n$, $l$, and $m$ as special cases. So Sommerfeld's approach was not so ad-hoc as this summary makes it look.

... model⁷

Bohr did not make use of the quantum number $m$; instead, he had $n$ slots at level $n$ ( $l=0,1,\ldots,n-1$), each capable of holding $2n$ electrons. Note that this is actually incorrect.

... vector⁸

We mean an ordinary 3-space vector, not a vector in a (complex) Hilbert space.

... system⁹

ignoring any motion of the proton, a good approximation

... publish¹⁰

Alfred Landé independently came up with the same half-integer trick. He did publish, and so attached his name to the Landé $g$-factor.

... together).¹¹

These lines are responsible for the color of sodium vapor lamps used on many highways.

... spectrum.¹²

The sodium D-lines provide an example. The quartet and the higher doublet both have $L=1$ and $S=\frac{1}{2}$; their $J$'s are $\frac{3}{2}$ and $\frac{1}{2}$.

... ``small''¹³

In particular cases, ``small'' and ``slowly'' can be given more definite meaning. For example, we may be able to write $E$ as a power series in something; $V$ might contain the second and higher order terms.