I think we can reconcile the passage in Gittin with the quote from Rashi as follows:
It seems that the key point from Rashi is whether the dispute is a matter of logical application or a historical fact. When one person said something, as in Rashi's case, there are no two possibilities that can be correct for what was said. Therefore, if there are multiple versions of what was said in a particular statement, to the exclusion of anything else, one version must of necessity be false.
In the case in Gittin truly parallel to this? The two parties there are not debating what someone said at a particular time. While it may seem that they are debating the historical factual reality of a particular event, I would argue that that may not be the case either. Consider the fact that neither R. Evyatar nor R. Yonatan were present at the event under discussion; nor do either of them assert any kind of tradition that the event happened in the particular way they are advocating. What reason, therefore would they have to stake out positions on something that happened long in the past, with no available eyewitnesses?
It would seem reasonable that rather than debating the factual reality they were debating the explanation of the Scriptural verse. The verse in question states וַתִּזְנֶה עָלָיו פִּילַגְשׁוֹ and the debate centers around what this phrase conveys. As Tosafot on the spot points out, one of the disputants interpreted וַתִּזְנֶה as relating to the same root as זנות, which would indicate some form of sexual impropriety, while his opponent interpreted it as relating to the root מזון, which would indicate something to do with food.
Thus, when R. Evyatar said that the husband found a fly in his food, he is actually saying that the verse indicates that he found a fly in his food, since it tells us that there was something involving מזון. And when R. Yonatan says that he found a hair in her genital area he is actually saying that the verse indicates that he found a hair in his genital area since it tells is that there was something relating to זנות.
The fact that these two conclusions seem to be at odds with each other does not mean that the factual conclusions were what was being debated. Indeed, this is precisely why the Talmud is able to say that both are true. Because they were merely discussing the implications of the verse, the Talmud is able to say that the verse in fact implied both of those things, and then elaborate on what the value of each of those things was in terms of the causation of the ensuing incident.
When read this way, we can actually resolve an additional difficulty. If this was merely a dispute about what event was the proximate cause of the incident of "pilegesh b'givah", how can the Talmud say that both opinions are correct when it then immediately explains that one opinion was not really correct (as it says that both events occurred but only one of them was actually the cause)? That does not seem to be sensible. However, once we view the dispute as being about Scriptural interpretation, the Talmud's explanation fits very nicely. Elijah says that both opinions are the words of God – this is true because the verse indeed does convey both meanings (and presumably deliberately so). When the Talmud then clarifies which one of the two events was the actual cause of the incident it is not casting inaccuracy on either of the two opinions; it is simply explaining that though the verse conveys that two different events happened, one of the events was actually forgivable while the other one led to the incident of "pilegesh b'givah". Seen this way, neither disputant was incorrect. They both provided correct interpretations of the verse, and both interpretations indeed corresponded to reality. The question of which of the two events actually caused the incident is a "fun fact", but it was not the primary discussion. Indeed, the Talmud does not offer any explanation as to how both events could have been the proximate cause – and there is no need for it to do so, as the "eilu v'eilu" is already accounted for with the Scriptural interpretations.
This fits back with Rashi in Ketubot as Rashi there says that when two people are discussing an interpretation, or applying one idea to another, or comparing cases, there is no necessity for one opinion to be objectively false. That is exactly what happened in Gittin.