‘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.