"He was ignorant of my other faults, else he would not have mentioned these alone."
The first quote this statement reminded me of was The Four Great References, which include,
In such a case, bhikkhus, the declaration of such a bhikkhu is neither to be received with approval nor with scorn. Without approval and without scorn, but carefully studying the sentences word by word, one should trace them in the Discourses and verify them by the Discipline. If they are neither traceable in the Discourses nor verifiable by the Discipline, one must conclude thus:
'Certainly, this is not the Blessed One's utterance; this has been misunderstood by that bhikkhu — or by that community, or by those elders, or by that elder.'
In that way, bhikkhus, you should reject it. But if the sentences concerned are traceable in the Discourses and verifiable by the Discipline, then one must conclude thus:
'Certainly, this is the Blessed One's utterance; this has been well understood by that bhikkhu — or by that community, or by those elders, or by that elder.'
And in that way, bhikkhus, you may accept it on the first, second, third, or fourth reference. These, bhikkhus, are the four great references for you to preserve.
In summary, IMO a statement like, "He was ignorant of my other faults", is, "Certainly, not the Blessed One's utterance" (i.e. the Buddha wouldn't say that).
Now it is maybe an example of a "confession" (e.g. "I behaved wrongly, I was muddle-headed") -- which can be found in Buddhism, perhaps more especially in the Vinaya than the Suttas (though also in at least one Sutta I can think of).
Instead, the view that "I have faults" might be a kind of statement (or view) which Buddhism hopes you might abandon -- perhaps the anatta (non-self) view might suggest something like, "A fault has arisen ... but it is not "mine" ... I see the disadvantage of it, and resolve to not make a habit of it."
I don't know, are you familiar with the Christian tradition? The statement you quoted seems to be a reaction to being spoken ill of (along the lines of "turn the other cheek"). Whereas I think the Buddhist doctrine is more to be unmoved or unreactive, "neither welcoming praise nor rebelling against disgrace or blame" (Lokavipatti Sutta). Also (continuing the comparison with Christianity) the Buddhist seems to me to be closer to Christ's doctrine in the Gospels (e.g. "be ye perfect"), than it is to the Apostles' doctrines in the Epistles (e.g. "we are sinners").
A close sutta, very apposite to your question, might be Akkosa Sutta: Insult (SN 7.2) -- what the Buddha taught when someone tried to insult him.
On the other hand perhaps it's right to be honest about whatever faults you do (or did) have (see also "confession" above). I don't recall suttas which illustrate that side of the teacher (or "good friend") relationship, but AN 5.123 describes caring for someone who is physically ill, and says that a patient who is accurate and honest about their symptoms is easy to care for.
Also, getting onto the topic of "I have other faults" sounds to me like maybe papañca, and the "thicket of views" which we're warned is associated with self-view and with "attending inappropriately" (if I may say so without wanting to insult your Stoicism).
"What ought one to say then as each hardship comes? I was practising for this, I was training for this."
A good example of that might be the Punna Sutta.
Perhaps one difference though, notable, is that Punna's replies are not very self-centered ... not, "I was training for this", but rather, "they are very civilised" ... and, so, maintaining a mind of good will (see also the "Brahmaviharas" which summarise the appropriate social or inter-personal attitudes).
I think that maybe you can find statements from the Buddhist tradition, which are close to what you have in mind.
For example Is That So? is an example of, "do not make excuses about what is said of you"; or The Taste of Banzo's Sword an example of, "If you want to improve, be content to be thought foolish and stupid".
But I wanted to point out that the doctrine of the Pali suttas is comparable to, but, not the same as those Stoic statements.