‘Square-integrable’ and ‘cuspidal’ are definitely not equivalent; the ~~former~~ latter representations are among the ~~latter~~ former, but do not exhaust them. To the best of my knowledge, ‘supercuspidal’ and ‘cuspidal’ are just the same concept with different etymologies behind them; and ‘absolutely cuspidal’ is meant to refer to representations that remain cuspidal upon extension of the ground field, hence is only interesting for representations in non-algebraically-closed fields. (For $p$-adic groups, the terminology ‘supercuspidal’ is used almost exclusively. I think one sees ‘cuspidal’ more in the global- or finite-field setting.)

I find it amusing that ‘very cuspidal’ (as used by Carayol, for example) is more restrictive than ‘supercuspidal’!

In the automorphic-forms settings, discrete-series representations are those that appear as direct summands of $L^2$ of our symmetric space. In this sense, they should be viewed as subrepresentations of the part of $L^2$ that decomposes ‘discretely’, as a direct sum, rather than continuously, as a direct integral. (In a measure-theoretic sense, the discrete series contains the atoms for the Plancherel measure. I think, but wouldn't swear to it, that Plancherel measure is absolutely continuous on the remainder of the tempered spectrum.) The cuspidal representations are those that appear in the space of $L^2$ functions that die off rapidly—in other words, that vanish at the cusps of a suitable compactification of the symmetric space. It is a theorem that they automatically appear as direct summands.