If you are going to try to quote Augustine, it is critically important you say
where he said these things. There are many false quotes that go around, which is why it's
important to show where they said it. As Abraham Lincoln wisely said: "Just because a quote is posted on the Internet does not mean it is true."
So you offer as quotes two images that give alleged quotes from Augustine, namely:
"Men are not Saved by Good Works, nor by the Free Determination of their Own Will, but by the Grace of God through Faith"
and
"God chooses us, not because we believe, but that we may believe."
No source is cited for these. Before we even get to whether Augustine said these things, it's not clear how these are relevant. You are holding them up, apparently, to try to support your statement of "But the issue at hand is your dislike of Calvinism and denying the security of the believer being secure forever in Christ. I don’t know which St Augustine you do read. But the St Augustine’s I read says exactly that". In other words, these are apparently supposed to support the claim he supported the doctrine of eternal security (often known colloquially as "once saved always saved"). Except I don't see how they're even relevant to that question.
But, regardless of their relevance, are these quotes accurate? I searched for these quotes online to see if someone else offered them. The first one you offer, "Men are not Saved by Good Works, nor by the Free Determination of their Own Will, but by the Grace of God through Faith"? Well, a search shows it's the title of a chapter in his work The Enchiridion (from the "Nicene and Post-Nicene Fathers" translation), which can be seen
here (also
here if one wishes to see the whole work at once, though it lacks the footnotes).
Sometimes these chapter titles aren't from the original author, and instead are added by a later editor or translator for reader convenience. To check into it, I tried to search up the Latin text in Patrologia Latina,
which is found in volume 40 (volume 6 of its volumes of Augustine), we can find it is there... sort of. It says:
"Non meritis, nec libero arbitrio reparari homines, sed gratia"
So we do see this here, meaning it wasn't added in the translation, though I suppose it's possible in these things it got added in collections; I'd have to look at manuscripts to be sure. But regardless, I have to question the translation of the above as "men are not saved by good works, nor by the free determination of their own will, but by the grace of God through faith". I hesitate to do this because I'm sure the person who translated this ("J.F. Shaw", whoever he was) presumably knew Latin a whole lot better than I do, but some of the translation choices seem off.
A fairly literal translation of the above Latin would be something like:
"Not [by] merits, nor [by] free will are restored men, but [by] grace."
Or in more natural English:
"Men are not restored by merits, nor by free will, but by grace."
Let's discuss the points where the translation seems to diverge from the above. "Meritis" (ablative plural form of meritum) is translated as "good works". This seems to me an odd choice. Meritum, deriving from the Latin verb mero ("to deserve") means merit (noun). Based on this dictionary entry, it seems it can refer to both merit in the sense of a kind act or merit in the sense of something that is deserved. If we go by the "kind act" definition, "good works" could work as a translation, but if Augustine was specifically referring to that he could have said "bona opera" (literally, "good works"). In fact, Augustine
does use a similar phrase in the actual chapter itself, "Quid enim
boni operatur perditus, nisi quantum fuerit a perditione liberatus", which the English translation renders as "For what
good work can a lost man perform, except so far as he has been delivered from perdition". Technically speaking the "boni operatur" is not "good works" but rather talking about the
good a man can
work, but this adjustment for sounding natural in English makes sense. At any rate, "good works" appears a stretch of a translation.
The translation of reparari as "saved" also seems rather odd. One could render it as "redeemed" (that's what Google Translate does, for the record, if you put this phrase there), but "saved" seems a step too far. The Latin word for saved is normally salvari/salvus. This is what is used in verses in the Latin version of the New Testament to refer to being saved, such as
Matthew 10:22 ("his salvus erit") or
Matthew 19:25 ("salvus esse"). Even more oddly, the exact same word is translated as "restored" in the text of the chapter itself, making it being "saved" in the title even more odd.
And lastly, "through faith" is not found
at all in the chapter title. The translator just added that in. Now, "through faith" is mentioned in the chapter itself (namely, when Paul is quoted), but it is not in the chapter title.
Perhaps the translator was working from a different manuscript or something, but compared to the one in Patrologia Latina, it looks like they made some questionable translation choices and in fact added words not even there. I admit I am far from an expert on Latin. Maybe there are very valid reasons for these apparent errors that I don't know about because I don't know enough, and I'm suffering from the Dunning Kruger effect. But it certainly
looks to me like the translation has been twisted somewhat.
As for the text of the chapter itself, the point Augustine is trying to make seems that one cannot restore themselves with God by works or their own free will, and that God has to free them from sin in order to do any of that. So this has nothing to do with eternal security, and if the point is to try to say this goes against Catholic doctrine, it doesn't seem to at all. These canons of the Council of Trent (Session 6) seem clear endorsements:
"If any one shall say, that man may be justified before God by his own works, whether done through the strength of human nature, or through the teaching of the law, without the divine grace through Jesus Christ; let him be anathema." (Canon 1)
"If any one shall say, that without the preventing inspiration of the Holy Ghost, and his help, man can believe, hope, love, or be penitent, as he ought, so that the grace of justification may be conferred upon him; let him be anathema." (Canon 3)
Or in Chapter 5 of the same session, it declares:
"...yet is he not able, without the Grace of God, by his own free will to move himself unto justice in His sight."
Thus, I don't see how Chapter 30 of the Enchiridion, when taken in its entirety rather than a possibly mistranslated title, goes against Catholic doctrine? The Council seems to be saying what Augustine said here.
As for the other quote of Augustine ("God chooses us, not because we believe, but that we may believe.") a search for that did not turn up anyone offering a source, but ChatGPT was able to point me to the source. (ChatGPT and other AI must be used cautiously and
always be verified, but are nevertheless a useful tool for finding things) This is Chapter 38 (or 29 in an alternate division) of On the Predestination of the Saints. Book 1 of that can be found
here in full or just the chapter
here if one wants the footnotes.
I did not look up the Latin on this one so I cannot vouch for translation quality, but again, how does this go against any Catholic doctrine? It seems in accord with what the Council of Trent said.
So it's unclear to me what these quotes are supposed to prove. As noted, they don't relate to eternal security to begin with (they really relate more to the question of predestination, which while related, is not the same thing), and even if the goal was to try to point to some other contradiction with Catholicism, they don't seem to?
tl;dr: The first Augustine quote is found in the English translation, but makes what appear to me questionable translation choices and a more accurate rendering appears to be "Men are not restored by merits, nor by free will, but by grace". The second one seems legitimate. However, neither really relate to eternal security, and neither (especially after the apparent translation errors are fixed) appear to go against Catholic doctrine.