Said quickly, solstices are rather perceptible while the equinox is a mental construction. Archeoastronomical evidence shows that neolithic people already had knowledge about the solsticial points on the horizon and eventually they had also some interest in the point that lies in the middle between them. When the sun rises from this point the duration of the day should be equal to that of the night. Measuring these time lengths with sufficient precision is actually a demanding practical task (and later it was shown that on this date the observable day is slightly longer). But idea is easy to conceive and there should be various mentions in preserved texts. References are what this question is about.
This is a partial answer, offering references relevant to ancient observation methods for equinoxes and solstices. References below cover only the near and middle East (recognizing that the question would also cover early knowledge from any period or civilization).
'Preserved [ancient] texts' with descriptions of how to detect and measure the times of equinox are rare. Fragments with indirect and partial references to this topic are perhaps a little more numerous.
It seems that the best that can be done in answer to the request for references, beyond some primary texts, is to give also some major studies about them, first because these often gather in some of the content of other and fragmentary older works, which will not otherwise appear, and then also because the primary texts are sometimes in need of explanation.
Primary texts for present purposes must practically include Ptolemy's Almagest, which has a good 1984 (annotated) translation by G J Toomer. There are online google reprint metadata about it, but as far as I know no online text.
In the absence of online text it may be of interest to add that measuring devices and methods relevant to equinoxes &c are described in text and accompanying notes at Toomer's p.61... (Book I.12. 'On the arc between the solstices') and p.133... (in part of Book III.1. 'On the length of the year').
Ptolemy describes two such measuring devices suitable for taking meridian altitudes or zenith distances of the sun around the times of solstice or equinox, and later a third device to indicate when there is an equinox.
Ptolemy first describes an armillary device with two rings, to be mounted vertically and in the N-S meridian plane, checking for alignment with a plumb- line for the vertical and a N-S meridian line on the ground. (He assumes his reader already knows how to locate and mark a meridian line.) The device has an outer ring, with graduation marks to indicate degrees from the zenith. An inner ring, sliding within the outer, is fitted with two exactly-opposite little marker-plates or -blocks, each with a pointer that grazes over the nearest graduation marks on the outer ring. At noon, the inner ring is adjusted by rotation, sliding within the outer, until the shadow of the forward marker cast by the sun falls exactly on the backward marker. Then the angular positions of the markers against the graduations can indicate the sun's meridian zenith-distance.
A second device is a 'plinthis', a plinth or block that stands on a horizontal pavement, and has a flat vertical face in the N-S meridian plane. A quadrant sector is marked on this vertical face, with its arc marked in degrees. The corner of the quadrant sector (center of the circle of which the arc is a part) is at top south. One radius extends horizontally back from the corner, towards north, the other radius passes vertically down from the corner. The quadrant is fitted with two pegs or markers: one is at the corner of the quadrant, the other vertically below it, on the circumference of the arc. The noonday sun throws a shadow from both markers. Ptolemy suggests locating more exactly where the shadow of the marker-peg at the corner falls on the graduated arc, by moving some object along the arc until it catches the mid-point of the shadow. The purpose of the lower marker peg is not explained in Ptolemy's text.
Ptolemy doesn't say much specifically about the equinox measurement, but does note that the moment of equinox is somewhat indeterminate. He mentions how the extremes of meridian zenith-distance of summer and winter define the latitude and the obliquity of the ecliptic (not using that exact expression).
Later he mentions a ring of rectangular cross-section mounted in the plane of
the equator to indicate an equinox when the sun (by change of declination)
passes though its plane. On such an occasion the inner surface of the ring
begins to get its light from above instead of below, or vice versa.
Such a ring had been mounted in Alexandria at the 'square Stoa', and its
somewhat irregular indications there had been referred to by Hipparchus.
Ptolemy noticed the practical shortcomings of two such rings in the
Alexandria of his own time, sometimes 'showing' two 'equinoxes' in a day.
He put the irregularities down to misalignment rather than to any irregular motion of the sun. It is now known that atmospheric refraction can also contribute to such errors.
Also mentioned by Ptolemy is the vertical gnomon. He says that the tip of the shadow is hard to discern/measure in winter.
There are instructive ancient commentaries on Ptolemy by Pappus and Theon, which likewise gather in some material from still older sources. They have been given modern translations (but so far as I have seen, only into French):--
A. Rome (ed.), Commentaires de Pappus et de Theon d'Alexandrie sur l'Almageste. Tome I. Pappus d'Alexandrie, Commentaire sur les livres 5 et 6 de l'Almageste. (Studi e Testi 54). Roma. (1931). Tome II. Theon d'Alexandrie, Commentaire sur les livres 1 et 2 de l'Almageste. (Studi e Testi 72). Citta del Vaticano, (1936). Tome III. Theon d'Alexandrie. Commentaire sur les livres 3 et 4 de l'Almageste. (Studi e Testi 106). Citta del Vaticano. (1943).
The Almagest texts are usefully read in conjunction with three further modern studies:
O Neugebauer's (1975) History of Ancient Mathematical Astronomy;
A Survey of the Almagest With Annotation and New Commentary by Alexander Jones (originally O Pedersen, 1974; new edition 2011);
and for the observational data and methods in the Almagest, also some important background and discussion in: J P Britton (1992) Models and precision : the quality of Ptolemy's observations and parameters.
Further material on Theon's commentary is at
A. Rome, 'Les observations d'equinoxes et de solstices dans le chapitre I du livre 3 du Commentaire sur l'Almageste par Theon d'Alexandrie'; [I] Annales de la Societe Scientifique de Bruxelles 57, ser.I, (1937), 213-236; [II] ibid. 58, ser.I, (1938), 6-26.
There is also a current research project for producing new editions of works not only by Ptolemy but related material by other authors, in Arabic and in Latin: the website is growing and already has much bibliographic material for searching related ancient sources:
'Ptolemaeus arabus et Latinus', see (https://ptolemaeus.badw.de).