Well, regardless of whether you're used to it or not, "triad chord" is not the proper term and you should get used to something else. "Triadic chord" is slightly better, but you should really just call one of these chords a "triad."
There are four kinds of triad: major, minor, diminished and augmented. You don't need to refer to scales to talk about triads, just the distance between the three notes. If there are four half steps between the bottom two notes (e.g. C–E) and three half steps between the top two notes (e.g. E–G), it's a major triad. If you reverse that (3 half steps on bottom like C–Eb and 4 on top like Eb–G) then it's a minor triad. I won't worry about the other two triadic qualities for this answer, but you can look them up easily.
In your question, you were mostly talking about the triads that happen naturally when you use only the notes of the C major scale—these are called the "diatonic triads." Three of them will automatically be major: C–E–G, F–A–C, and G–B–D. In fact, in any major key, the triads that you can build on the first, fourth and fifth scale degrees will be major triads. In C major, three of the triads will naturally be minor: D–F–A, E–G–B and A–C–E. In fact, in any major key, the triads you can build on the second, third and sixth scale degrees will always be minor. (The chord built on the seventh scale degree, B–D–F in C major, will be diminished)
But to answer your question about tuning triads, it's easier to focus on a major and a minor triad with the same root. As you point out in your question, C–Eb–G would be a minor triad and C–E–G would be major. If you play these notes exactly as a standard tuner says they should be, the distance between the third of the minor triad (Eb) and the third of the major triad (E) is 100 cents, so micro tunings like you're talking about won't change the fundamental chord quality. The problem is that both minor and major triads tuned exactly according to the tuner (this is called equal temperament) don't sound as pure and powerful. This is because of something called the harmonic overtone series, and you can find tons of information here by searching for that term and the phrase "just intonation." You're right that raising the third of a minor triad by about 15 cents will make the minor triad more purely "in tune." However, it's just as important to lower the middle note of a major triad by 15 cents. So, to get a pure C major chord, you would play C–E–G with the E about 15 cents lower according to a tuner, and to play a C minor triad you would play C–Eb–G with the Eb about 15 cents higher. Although you're right that this makes the middle notes of the two triads closer together, it's not nearly enough to make the major triad become minor or vice versa. There's no set number, but I'd say you'd have to raise the third of a minor triad at least 60 cents before it starts truly sounding like a major triad. But the beauty of this is that you start to learn just how much range and nuance there is between notes that are supposedly "right next to each other."