Did different Greek mythology sources ever explicitly acknowledge the contradictions between themselves?
Greek mythology is full of different accounts of the same exact facts, based on which source you use.
As a random example, here's the assorted sourced details on Prometheus' parentage from Wikipedia:
In Hesiod's Works and Days Prometheus is addressed as "son of Iapetus", and no mother is named. However, in Hesiod's Theogony, Clymene is listed as Iapetus' wife and the mother of Prometheus. In Aeschylus's play Prometheus Bound, Prometheus is son of the goddess Themis with no father named (but still with at least Atlas as a brother). However, in Horace's Odes, in Ode 1.3 Horace describes how "audax Iapeti genus/ Ignem fraude mala gentibus intulit"; "The bold offspring of Iapetus [i.e. Prometheus]/ brought fire to peoples by wicked deceit".
This is by no means the only point of discrepancy between various Greek mythological sources. I can probably stick my finger into a random Mythology.SE answer and find one (e.g., a random Q&A I read today, about Gorgon's sight being non-lethal because of the shield, vs. because she was asleep and there was no shield-mirror-gazing at all).
Did any of the well known sources ever explicitly acknowledge the differences between their own version of the myth details, and another source's version? I'd prefer a "bulk" acknowledgement (e.g. the fact that many details differ), but one only highlighting difference of a specific detail is sufficient.
They indeed did, and they highlighted it in different ways. One way was to negate the earlier myth. I gave one example in a previous thread. In Pindar's Olympians ode 1, he references the fact that others before him have a version of a myth that is incompatible with his beliefs, namely that Tantalus would kill his son Pelops and feed him to the gods. Instead he creates a story of jealous neighbors who propagate the false myth; really, Pelops was just on Olympus with Poseidon all along!
Another way to do it is to just ignore the discrepancy, because the discrepancy is not the main point. Hesiod's Works and Days has a good example of this.
First, he includes a self-correction:
So, after all, there was not one kind of Strife alone, but all over the earth there are two.
In Theogony 225, Hesiod originally listed the goddess Strife (Eris in Greek), who was born of Night (Nyx). Because the point of the Works and Days is different from Theogony, in that he wants to promote the labor of men and hard work, he invents another Strife to use for didactic purposes.
But the real inconsistency lies further in. In Works and Days 42-104, Hesiod narrates the fall of man through the story of Pandora. After Prometheus stole fire from Olympus and gave it to men, Zeus, angry that men might not now work as hard, devised a plan to give them woman. He made the first woman, Pandora, gave her intense curiosity, and then gave her to Epimetheus, the brother of Prometheus. He then gave Epimetheus a box (nowadays called a jar) filled with various evils. Her curiosity got the better of her, and she opening the box let out all the evils to plague mankind, making it worse than before.
However, he thinks that maybe his story didn't quite get across:
So is there no way to escape the will of Zeus. Or if you will, I will sum you up another tale well and skilfully—and do you lay it up in your heart,—how the gods and mortal men sprang from one source.
Now (109-179) Hesiod narrates the "races of men," in which mortals are created by the gods in several different metals: gold first, and blessed are they; then silver, but they were full of hubris; then bronze, but they were too bloodthirsty; then the "Age of Heroes", i.e. the Trojan and Theban material; lastly the iron age, which is the current age, and was made degenerate. There is no Pandora, there is no Promethean fire, and that doesn't seem to bother him much. Perhaps these might be closer to parables, but Greek writers believed in them in earnest.
Later authors, especially late ones, tended to accept some and expect variation among reports. Diodorus Siculus is a good example of this one. When discussing the famous Calydonian Boar hunt, he first gives a brief overview of the accepted "facts":
The facts are these: Once when Oeneus had an excellent crop of grain, he offered sacrifices to the other gods, but neglected Artemis alone; and angered at him for this the goddess sent forth against him the famous Calydonian boar, a creature of enormous size. 3 This animal harried the neighbouring land and damaged the farms; whereupon Meleager, the son of Oeneus, being then in the bloom of youth and excelling in strength and in courage, took along with himself many of the bravest men and set out to hunt the beast. Meleager was the first to plunge his javelin into it and by general agreement was accorded the reward of valour, which consisted of the skin of the animal. 4 But Atalantê, the daughter of Schoeneus, participated in the hunt, and since Meleager was enamoured of her, he relinquished in her favour the skin and the praise for the greatest bravery. The sons of Thestius, however, who had also joined in the hunt, were angered at what he had done, since he had honoured a stranger woman above them and set kinship aside. Consequently, setting at naught the award which Meleager had made, they lay in wait for Atalantê, and falling upon her as she returned to Arcadia took from her the skin. 5 Meleager, however, was deeply incensed both because of the love which he bore Atalantê and because of the dishonour shown her, and espoused the cause of Atalantê. And first of all he urged the robbers to return to the woman the meed of valour which he had given her; and when they paid no heed to him he slew them, although they were brothers of Althaea. Consequently Althaea, overcome with anguish at the slaying of the men of her own blood, uttered a curse in which she demanded the death of Meleager; and the immortals, so the account runs, gave heed to her and made an end of his life.
He then immediately follows it up with an account given by "certain writers of myth"
But certain writers of myths give the following account: — At the time of the birth of Meleager the Fates stood over Althaea in her sleep and said to her that her son Meleager would die at the moment when the brand in the fire had been consumed. Consequently, when she had given birth, she believed that the safety of her child depended upon the preservation of the brand and so she guarded the brand with every care. 7 Afterward, however, being deeply incensed at the murder of her brothers, she burned the brand and so made herself the cause of the death of Meleager; but as time went on she grieved more and more over what she had done and finally made an end of her life by hanging.
And that's just it! It doesn't seem to bother Diodorus at all, and it likely didn't bother very many Greeks.
In a similar way, the accounts of Electra in Aeschylus' Oresteia, Sophocles' Electra, and both Euripides' Electra and his Orestes all differ greatly from each other, depending on what they want to emphasize. This was not only acceptable, but encouraged, as sticking too close to someone else's interpretation was considered bad-form, as some comedians were accused of doing; check out Scott McGill's Plagiarism in Latin Literature (2012) pp. 130-131 (earlier pages are unfortunately hidden for Google Books, but the discussion goes on a bit before and after if you want to find a real copy) for an early Roman accusation against Plautus and Terence.