While Etymonline isn't particularly reliable (and has a strong aversion to citing any sources), this line almost has it right:
The Latin word translates Greek apolambanomene.
It's a calqued Greek word, though not the one Etymonline claims.
As far as I can determine, Apollonius of Perga is the one who first introduced the term, in his enormous treatise on conic sections. He uses the word ἀποτεμνομένη ("amount-being-cut-off") for what we might now call an "X-coordinate".
For example, here's Proposition 20 from Book 1:
Or, in a more legible typeface:
Ἐὰν ἐν παραβολῇ ἀπὸ τῆς τομῆς καταχθῶσι δύο εὐθεῖαι ἐπὶ τὴν διάμετρον τεταγμένως, ἔσται ὡς τὰ ἀπ' αὐτῶν τετράγωνα πρὸς ἄλληλα, οὕτως αἱ ἀποτεμνόμεναι ὑπ' αὐτῶν ἀπὸ τῆς διαμέτρου πρὸς τῇ κορυφῇ τῆς τομῆς.
If, in a parabola, two straight lines are dropped perpendicular, from the cut [i.e. the curve of the parabola] to the diameter, then squares built on them [on these straight lines] will be to each other, just as the parts sliced off from the diameter by this cutting, from them [the lines] to the vertex.
Or, to modernize it a little bit:
If you take any parabola, and mark two points on it, then the ratio of the squares of the vertical distances of those points to the diameter, is equal to the ratio of the horizontal distances of those points from the vertex (along the diameter).
(The "diameter" of a parabola is a bit of an archaic term: it's the line parallel to the directrix passing through the vertex. For the parabola y=ax², the "diameter" is the X-axis.)
In other words, he's figured out the equation of a parabola: nowadays, we'd put this more simply as "y=ax², for a constant a". What we might now call a "Y-coordinate", he calls the length of a perpendicular dropped from a point to the diameter. And what we might now call an "X-coordinate", he calls an ἀποτεμνομένη, an "amount-being-cut-off"—because to him, the X-coordinate is how much of the diameter we've marked off by dropping that perpendicular.
Or, to put it another way: Apollonius is marking a point on the parabola, then dropping a perpendicular from that point to the X-axis. This effectively slices the X-axis apart; the sliced-off bit that lies between the origin (in this case the vertex) and our perpendicular is the ἀποτεμνομένη. Earlier, he uses the word τομή "cut" for the curve of the parabola itself: you might imagine all these lines and marks being engraved into a piece of material, so that every line we draw is indeed a "cut".
When Apollonius's works were translated into Latin, ἀποτέμνω was calqued to abscindō, "to cut off". But Latin doesn't have a present passive participle like Greek does. So some translators used a finite verb form, like this one (from Apollonii Pergæi Conicorum libri octo, et Sereni Antissensis De sectione cylindri & coni libri duo, Oxford, 1710):
Si in parabola duæ rectæ à sectione ad diametrum ordinatim applicentur: ut eorum quadrata inter sese, ita erunt et rectæ, quæ ab ipsis ex diametro ad verticem abscinduntur.
If two straight lines are drawn in a parabola, going from the cut [the curve of the parabola] perpendicularly down to the diameter, then just as is the ratio of their squares, so will be the ratio of the straight lines which are cut away from the diameter, running from them [the vertical lines] to the vertex.
And others used a perfect passive participle, like this one (from Apollonii Pergaei quae graece exstant cum commentariis antiquis: Praefatio. Conicorum lib. I-III, translated by I.L. Heiberg):
Si in parabola a sectione duae rectae ad diametrum ordinate ducuntur, erunt, ut quadrata earum inter se, ita rectae ab iis e diametro ad verticem sectionis abscisae [sic].
If, in a parabola, two straight lines are led [i.e. drawn] from the cut to the diameter, then it shall be: just as their squares, thus the straight lines from them to the vertex, cut from the diameter.
Giving us the now-familiar abscissa: the "amount that has been cut off". This one could be used as a noun, unlike the finite verb version, so it's the one that stuck.
