abovedisplayskip vs abovedisplayshortskip

Others have well described when the "short" skips are selected; as to why, it's easy: the purpose is avoiding large gaps between text and a centered equation.

How to avoid unbalanced spacing before and after equations? The answer is simple: use amsmath that provides many environments in order to accommodate multiple equations or alignments. For example, your case study should be input as

\begin{gather}
u_{i,t} = u_{i,t-1} + v_t \\
u_{i,t-1} = u_{i,t-2} + v_{t-1} \\
\Delta u_{it} = u_{i,t} - u_{i,t-1} = u_{i,t-1} + v_t - u_{i,t-1} =
v_t
\end{gather}

which avoids the "unbalancedness" and also prevents bad page breaks. By default amsmath will never allow a display environment to be broken across pages, but it's possible to override locally this behavior at final revision time with a suitably placed \displaybreak command.

The short versions are used when the line above or below (as appropriate) is 'short'. Now, that leaves the question of what is 'short'. To quote the excellent TeX by Topic

The ‘short’ variants of the glue are taken if there is no \leqno left equation number, and if the last line of the paragraph above the display is short enough for the display to be raised a bit without coming too close to that line. In order to decide this, the effective width of the preceding line is saved in \predisplaysize. This value is calculated immediately after the opening $$ of the display has been scanned, together with the \displaywidth and \displayindent explained above.

Thus the short skips are used where there is a visual 'overlap' between the paragraph text and the math mode material.

The short variants gets inserted when the previous/preceding paragraph is short, to compensate for the (optical) whitespace.

To paraphrase The TeXbook:

Let z, s, and p be the current values of \displaywidth, \displayindent, and \predisplaysize.

...

TEX computes a displacement d, to be used later when positioning box b,

...

If d + s ≤ p, or if there was a left equation number (\leqno), TeX sets g_a and g_b to glue items specified by the parameters \abovedisplayskip and \belowdisplayskip; respectively; otherwise, g_a and g_b become glue items corresponding to \abovedisplayshortskip and \belowdisplayshortskip. [Translation: If the predisplaysize is short enough so that it doesn't overlap the displayed formula, the glue above and below the display will be "short" by comparison with the glue that is used when there is an overlap.]