The triad built on the 7th scale degree is indeed a diminished triad, both intervals are minor thirds. In the key of C, that triad is the notes B, D, and F. There's a minor third from B to D, and another from D to F. So, the vii chord (if you're only going to the triad) is a diminished triad.
However, if you go to the four-note chord, then the next note diatonically is A. That's a MAJOR third from the F (the five of the chord), so the seventh chord built on that scale degree is NOT a diminished chord. It's a minor 7 b5, or a half-diminished chord.
A full diminished 7th chord would be all minor thirds. Using B as our starting point, a Bdim is B, half-step to D, half-step to F, half-step to Ab. But Ab isn't in the key of C, so a full diminished chord is not diatonic to the key of C.
This is only one of the reasons I find it so very critical to understand stacking thirds and harmonizing the diatonic major scale. It really clarifies WHY things are the way they are. I find it silly to try to "teach" by having folks memorize stuff. It's MUCH better to help a student understand WHY the I and IV are major 7, the ii, iii, and vi are minor 7, the V is a dominant 7, and the vii is minor 7 b5 than it is to have them memorize those things. It's the difference between teaching a man to fish, and giving him a fish.
So, how do you harmonize the major scale? Do it on paper (it works GREAT on staff paper if you read standard notation). But just using the letters gets the point across. So, get a sheet of paper and a pencil, I'll wait....
Write out the scale once from root to root.
C D E F G A B C
Now, above that write it out again, only starting on the third...
E F G A B C D E
C D E F G A B C
Then repeat it again, moving up another third...
G A B C D E F G
E F G A B C D E
C D E F G A B C
Those are the triad diatonic to the key of C. Analyze them. Figure out WHY the first stack of C, E, G is a major chord, just like the IV (F), and the V. Understand WHY the ii, iii, and vi are minors.
Then go one third higher to derive the 7th chords...
B C D E F G A B
G A B C D E F G
E F G A B C D E
C D E F G A B C
Now you can determine clearly WHY the V chord is different from the I and IV, why the vii is different from the ii, iii, and vi.
Plus you start to see the connections between chords- why the Amin7 can substitute for the C chord, how Dmin7 and Fmaj relate, etc. And this is the start to the way great jazz soloists think. Miles, Bird, Dizzy, etc. weren't thinking modes at all. They were thinking intervals stacked on top of intervals. It's the way Larry Carlton approaches guitar too.
John