Be Careful What You Imbibe!
From what little reading I've done on the subject of the "Two Logoi," particularly here, but also from my reading and studying the New Testament, particularly Colossians, I am guessing the second-century apologists had imbibed deeply of the philosophy of the Stoics as it was passed down to them over the centuries. In an attempt to harmonize Judaeo-Christian doctrines with the worldview of the Stoics, the apologists were in effect compromising unnecessarily the truth handed down to them from the original apostles (and later, Paul) via both oral and textual means.
In other words, in their attempt to accommodate "worldly philosophy," and for whatever reasons (e.g., saving face with their non- or anti-Christian peers), they abandoned "sound doctrine" (see 1 Timothy 6:3; 2 Timothy 4:3; Titus 1:9; Titus 2:1) in favor of what the apostle Paul would call
". . . philosophy and empty deception, according to the tradition of men, according to the elementary principles of the world, rather than according to Christ" (Colossians 2:8 NAS).
First Principles Are Critically Important
One of the most basic and "elementary principles of the world" according to Stoics and neo-Stoics alike was logos/λόγος, which to them was God with a capital G. As Copan and Lutwak put it, God to the Stoics was "the cosmos and all that is in it" (p.81, in The Gospel in the Marketplace of Ideas; IVP Academic, 2014). To coin a word, the Stoics were cosmotheists, much like the late Carl Sagan, to whom the cosmos was "all there was, all there is, and all there ever will be." In fairness to the Stoics, however, Sagan was a dyed in the wool materialist, whereas the Stoics were pure rationalists. To them, the cosmos was Rationality writ large. When we die, the Stoics thought, that spark of divine rationality would simply be absorbed into the Logos of the universe.
By both nature and nurture, all people, the Stoics believed, acquired the divine spark of logos in two ways: first, by simply being born a human being (and it did not hurt to be born a Greek!); and second, by being socialized and educated by the paidea/παιδεία, or in today's terms, the soft-and hard sciences.
First Principles Inform the Educational Process
According to Kamesar (linked to, above), the two Logoi to which you refer in your question corresponded to two of the key academic disciplines in the Paidea; namely, rhetoric and dialectic. The former was the study of the art of public speaking, so that whether in deliberative, forensic, or epideictic settings, an Athenian would acquit himself well, as befitting a valued and virtuous member of the polis. In other words, whether in the Agora (particularly at public occasions such as the funeral of a respected citizen), the Areopagus (i.e., the “Supreme Court” of Athens), or the Senate (where the body politic would debate issues pertaining to public life), an Athenian was expected to be a man of words.
Dialectic, on the other hand, was the art of thinking and reasoning in ways which were sound, philosophically, logically, and ethically. Dialectic was the "counterpart" of rhetoric, according to Aristotle. In order to speak well, one must first think well. This internal aspect of "pure" thought is what we today call intrapersonal communication or ideation or meta-thinking or philosophizing, to mention just a few. Only when this internal aspect of the Logos (viz., endiathetos) becomes refined according to the dictates of philosophy as mediated by education and especially the discipline of intelligent discussion with one's peers and one's "betters," will a rhetor have anything truly worthwhile to say (that's prodiathetos).
In short, in the Greek mind, rhetoric would be identified with prophorikos, and dialectic with endiathetos. The two were obviously linked, and each could potentially enhance the other, but only if the linkage were forged in virtue. If not, a trained speaker could sink to being a tricky speaker, like one of the Sophists, whom Plato and Aristotle vilified, some of whose stock in trade were “fallacious reasoning, intellectual charlatanism and moral unscrupulousness” (IEP).
In Democratic Greece, sophists were itinerant teachers who "in return for a fee . . . [would offer] young wealthy Greek men an education in aretē (virtue or excellence), thereby attaining wealth and fame" (ibid.). Some were ethical teachers simply with an entrepreneurial bent, while others, from the perspective of "the ends justify the means," were not averse to cutting corners vis-à-vis virtue.
An Alternative to False Teaching
In contrast to some sophists, the apostle Paul's modus operandi as God's chosen "apostle to the Gentiles" was refreshingly different, as he indicated in his second letter to the Corinthians,
"But we renounce the hidden things of dishonesty, not walking in craftiness nor adulterating the word of God: but by manifestation of the truth commending ourselves to every man’s conscience, in the sight of God" (4:2 RHE).
Moreover, Paul described his approach to preaching among the Corinthians as follows:
"And my speech and my preaching were not in persuasive words of wisdom, but in demonstration of the Spirit and of power" (1 Corinthians 2:4 ASV).
True enough, on occasion Paul did use his academic training in philosophy and rhetoric to reach out to an audience composed of worldly-wise Stoics and Epicureans (e.g., Acts 17:22 ff.), but then Paul was known for becoming all things to all men that by all [ethical] means he might win some (a slight paraphrase of 1 Corinthians 9:22).
In Conclusion: My Pre-suppositional Stance
I suggest that from a conservative and Evangelical perspective the apologists to whom you refer were false teachers, much like the Gnostics of the apostle Paul's and the apostle John's day. They were compromisers who, for whatever reasons, seemed to elevate the Logos of Rationality/Cosmos above even God the Father and God the Son, implying that the Logos somehow antedated both the Father and Son, which borders on blasphemy. In effect, these false teachers elevated some thing above God, a clear violation of the First Commandment. Moreover, their teaching conflicts with the clear teaching of Jesus, who said,
"'But when the Comforter is come, whom I will send unto you from the Father, even the Spirit of truth, which proceedeth from the Father, he shall bear witness of me [, the Logos]'" (John 15:26).
As you point out in your question, the Triune God is rational, and to be rational is a crucial characteristic of personhood. God's rationality, however, resides in His personhood eternally. Rationality has no independent existence apart from God. It is therefore one of the infinite attributes of divinity (related to omniscience) which God has placed to a finite degree in us, His image bearers.
God is not only rational, but He is also--and more significantly, I would add--relational. That relational nature is perhaps the key attribute which makes Him more than a principle, more than an impersonal demi-urge, and more than a mere abstraction into which all humanity will one day be absorbed. No, the eternal God and Father as revealed in all His fullness in the person of Jesus the Son, is Light, and He is Love:
"For it was the Father's good pleasure for all the fullness to dwell in Him . . ." (Colossians 1:19 NAS).
"For in Him all the fullness of Deity dwells in bodily form . . ." (Colossians 2:9 NAS).
"This is the message we have heard from Him and announce to you, that God is Light, and in Him there is no darkness at all" (1 John 1:5).
"The one who does not love does not know God, for God is love" (1 John 4:8 NAS).
"We have come to know and have believed the love which God has for us. God is love, and the one who abides in love abides in God, and God abides in him" (1 John 4:16 NAS).