I agree, it's a bit sloppy, but in the end the narrator is stating exactly what you want. Namely, that the memory of the weapon came from Rand, and not from LTT, as most memories of old things do. There are mutiple instances of BS making it clear that the reader is gaining insight into Rand's personal thoughts. He uses both italics, and literally says it as well. I am the Dragon Reborn, Rand thought. I break all covenants and vows. How odd, that they should find this now, he thought, and make.... Light! he thought. I'm losing control.

For the rest, I see it as the narrator commenting from Rand's POV, giving insight into Rand's thoughts but not voicing them directly. For example, "How did the questioning go, Cadsuane Sedai?" he asked in a more moderate tone. This is not Rand's direct thought, that he asked in a moderate tone, it's the narrator letting us know what's happening. Or Cadsuane sipped her wine. Rand isn't directly thinking this, it's the observation from his POV, that the narrator is letting us in on. Or "This isn't art, Cadsuane," Rand said dryly. Rand is not thinking to himself "I said that dryly", it's the narrator letting us know.

So, we have Rand's direct thoughts....How odd, that they should find this now, he thought, and make a gift of it to me, completely unaware of what they were holding... which was followed immediately by the narrator informing us that Rand had taken to wearing the sword immediately (certainly Rand is not thinking to himself at this point "I began wearing the sword immediately" ), that if felt right in his hands, that he had told nobody he recognized the sword, and that the recognition came from Rand, and not LTT as one would expect given the "centuries old" bit and given that LTT is the primary source of much knowledge for Rand.