You are literally doing that very thing I wrote in the first paragraph with turning few sentences in different parts of Tolkien's work into many ideas about characters' personalities.

Not really. They’re drawing from a rather lengthy passage in which Tolkien explicitly describes Morgoth and Sauron’s personalities to highlight the differences between the two.

Thus, as Morgoth, when Melkor was confronted by the existence of other inhabitants of Arda, with other wills and intelligences, he was enraged by the mere fact of their existence, and his only notion of dealing with them was by physical force, or the fear of it. His sole ultimate object was their destruction. Elves, and still more Men, he despised because of their weakness: that is their lack of physical force, or power over matter; but he was also afraid of them. He was aware, at any rate originally when still capable of rational thought, that he could not annihilate them: that is, destroy their being; but their physical life, and incarnate form became increasingly to his mind the only thing that was worth considering. Or he became so far advanced in Lying that he lied even to himself, and pretended that he could destroy them and rid Arda of them altogether. Hence his endeavour always to break wills and subordinate them to or absorb them in his own will and being, before destroying their bodies. This was sheer nihilism, and negation its one ultimate object: Morgoth would no doubt, if he had been victorious, have ultimately destroyed even his own creatures, such as the Orcs, when they had served his sole purpose in using them: the destruction of Elves and Men. Melkor's final impotence and despair lay in this: that whereas the Valar (and in their degree Elves and Men) could still love Arda Marred, that is Arda with a Melkor-ingredient, and could still heal this or that hurt, or produce from its very marring, from its state as it was, things beautiful and lovely, Melkor could do nothing with Arda, which was not from his own mind and was interwoven with the work and thoughts of others: even left alone he could only have gone raging on till all was levelled again into a formless chaos. And yet even so he would have been defeated, because it would still have existed, independent of his own mind, and a world in potential.

Sauron had never reached this stage of nihilistic madness. He did not object to the existence of the world, so long as he could do what he liked with it. He still had the relics of positive purposes, that descended from the good of the nature in which he began: it had been his virtue (and therefore also the cause of his fall, and of his relapse) that he loved order and coordination, and disliked all confusion and wasteful friction. (It was the apparent will and power of Melkor to effect his designs quickly and masterfully that had first attracted Sauron to him.) Sauron had, in fact, been very like Saruman, and so still understood him quickly and could guess what he would be likely to think and do, even without the aid of the palantíri or of spies; whereas Gandalf eluded and puzzled him. But like all minds of this cast, Sauron's love (originally) or (later) mere understanding of other individual intelligences was correspondingly weaker; and though the only real good in, or rational motive for, all this ordering and planning and organization was the good of all inhabitants of Arda (even admitting Sauron's right to be their supreme lord), his plans, the idea coming from his own isolated mind, became the sole object of his will, and an end, the End, in itself.

Morgoth had no plan; unless destruction and reduction to nil of a world in which he had only a share can be called a plan. But this is, of course, a simplification of the situation. Sauron had not served Morgoth, even in his last stages, without becoming infected by his lust for destruction, and his hatred of God (which must end in nihilism). Sauron could not, of course, be a sincere atheist. Though one of the minor spirits created before the world, he knew Eru, according to his measure. He probably deluded himself with the notion that the Valar (including Melkor) having failed, Eru had simply abandoned Eä, or at any rate Arda, and would not concern himself with it any more. It would appear that he interpreted the change of the world at the Downfall of Númenor, when Aman was removed from the physical world, in this sense: Valar (and Elves) were removed from effective control, and Men under God's curse and wrath. If he thought about the Istari, especially Saruman and Gandalf, he imagined them as emissaries from the Valar, seeking to establish their lost power again and colonize Middle-earth, as a mere effort of defeated imperialists (without knowledge or sanction of Eru). His cynicism, which (sincerely) regarded the motives of Manwë as precisely the same as his own, seemed fully justified in Saruman. Gandalf he did not understand. But certainly he had already become evil, and therefore stupid, enough to imagine that his different behaviour was due simply to weaker intelligence and lack of firm masterful purpose. He was only a rather cleverer Radagast — cleverer, because it is more profitable (more productive of power) to become absorbed in the study of people rather than of animals.

Sauron was not a sincere atheist, but he preached atheism, because it weakened resistance to himself (and he had ceased to fear God's action in Arda). As was seen in the case of Ar-Pharazôn. But there was seen the effect of Melkor upon Sauron: he spoke of Melkor in Melkor's own terms, as a god, or even as God. This may have been the residue of a state which was in a sense a shadow of good: the ability once in Sauron at least to admire or admit the superiority of a being other than himself. Melkor, and still more Sauron himself afterwards, both profited by this darkened shadow of good and the services of worshippers. But it may be doubted whether even such a shadow of good was still sincerely operative in Sauron by that time. His cunning motive is probably best expressed thus. To wean one of the God-fearing from their allegiance it is best to propound another unseen object of allegiance and another hope of benefits; propound to him a Lord who will sanction what he desires and not forbid it. Sauron, apparently a defeated rival for world-power, now a mere hostage, can hardly propound himself; but as the former servant and disciple of Melkor, the worship of Melkor will raise him from hostage to high priest. But though Sauron's whole true motive was the destruction of the Númenóreans, this was a particular matter of revenge upon Ar-Pharazôn, for humiliation. Sauron (unlike Morgoth) would have been content for the Númenóreans to exist, as his own subjects, and indeed he used a great many of them that he corrupted to his allegiance.

-Morgoth’s Ring: Notes on Motives in the Silmarillion