You must be careful, when reading a play, not to be too quick to look for "certain" proof of a character's motivation. You can certainly find all the evidence you might need for confirmation of things a character says, but, since all the reader of a script is given is what that character says (not the most reliable source of accuracy) and what other characters say about him/her (an even more unreliable source of accuracy), a reader is destined never to know for sure what motivates a character. This is why, if you, for example, Googled "Iago," you would get nearly as many opinions as entries about what his motivates his action in the play.
It is true that, in the long speech that you cite, Iago tells the story of being passed over for promotion by Othello in favor of Cassio. But, just after he has this long speech, he also tells Roderigo that "there's no remedy" for his being passed over, "'tis the curse of service" (seeming not to blame anyone, but to rather blame the way the military works).
And then Iago goes further in explaining that he only "follows" Othello to "serve [his] turn upon him." Here, he explains to Roderigo that he is not one of those "yes" men that do whatever they must for their superiors in order to win favor. This point of view, as he notes in talking about how the military operates, would hardly win him promotion, whatever else might take place.
But the most telling lines (57-64) come just after this. He says:
Were I the Moor, I would not be Iago:
By following him, I follow but myself.
Heaven is my judge, not I for love and duty,
But seeming so, for my own peculiar end.
For when my outward action does demonstrate
The native act and figure of my heart,
. . .'tis not long after,
But I will wear my heart upon my sleeve.
. . .I am not what I am.
These lines reveal a very important warning from Iago -- a warning to Roderigo, to all the characters in the play (though they don't hear it), and also to the audience. Iago is stating quite plainly here that he intends never to show his true face, that he will never, in effect, "wear [his] heart upon his sleeve." His own end, his desired outcome, is, he says, "peculiar," which suggests that it bears no logic, and, therefore couldn't be figured out. But most importantly, he warns everyone that he is not what he appears to be.
So, in brief, since Iago has confessed that he cannot be trusted in action or word to represent his own true feelings, then that means that we, the readers/audience, cannot know his true motivation or if his story of any event is accurate.
For more on Iago and truth-telling and Act I, scene i, please follow the links below.