Well, it's still a diminished chord. By definition. Believe me, I'm not making any of this up. But that's not the whole story. Again, it's a matter of context. There's no way of explaining this stuff briefly (would that there were), but if you follow me all the way here, I think you'll find the ultimate answer satisfactory.
Any group of three or more notes, composed of a root, 3rd, and 5th is a chord. A chord frequently has more notes than just three (I'm not talking about added-note chords like 6ths, 7ths, 9ths, 11ths, and 13ths here, that would just add needless complication, but suffice it to say that added note chords usually start with the basic triad [root, 3rd, and 5th] then go from there). For example, A first position C chord on the guitar. The notes, beginning with the fifth string and going up, are C E G (the basic triad) C (doubling the root) and E (doubling the 3rd). A full symphony orchestra may be playing hundreds of notes at any given moment, but it you eliminate all of the doublings, you usually wind up with three basic notes -- a triad. With some modern music, e.g. twelve-tone, all bets are off!
Both the major and minor chords contain an interval of a perfect 5th. It's the note between the root and the 5th that make the difference. A major chord has an interval of a major 3rd on the bottom and an minor 3rd on top. A minor chord has a minor 3rd on the bottom and a major 3rd on the top. Now -- a diminished chord has a minor 3rd on the bottom with another minor 3rd on the top. The result of this is that it does not contain a perfect 5th. It contains a diminished 5th. That's where it gets it's name. All it requires to identify it as a diminished chord is the basic triad -- three notes. You can say with certainty that in the key of C, the root of the diminished triad is , the 3rd is D, and the 5th (in this instance, diminished rather than perfect) is F. You form the Dominant 7th chord either by adding an F to the G major> triad, or adding a G to the B diminished triad. If you add a G# to the B diminished triad, you are adding a note that does not belong in the key of C.
But -- if you do add a G# to a B diminished triad, no one will smite you hip and thigh and cast you hence. In fact, you have a very useful chord. This is the configuration that is commonly called a "diminished chord."
In a way, a diminished chord (G# B D F) is neither fish nor fowl. It has a very vague identity. Follow me here, because this gets cute:
If you start with a Dominant 7th chord (say a G7), you instantly know what key you are in -- or, least what key you are heading for: C. The 3rd of a Dominant 7th (in this case, a B) is the "leading tone" of the key it's in, so you know it's pointing directly at the C chord. At the same time, the F wants to drop a half-step to an E. The ear wants to hear a C chord. It may not actually go there, but that's the kind of tease that keeps a piece of music going. I'm sure you're familiar with songs, e.g., in the key of C, where you go to a D7 then to a G, then to a G7, and back to a C again. That's a mini-modulation. For a second there, you were in the key of G before you changed it to a G7 and returned to the original key.
With our four note diminished chord, all of the intervals are dead even. It is a pile of minor thirds. If you invert any of the notes (take the bottom note and put it on top) you still have a pile of minor thirds. In addition, all of the 5ths are diminished 5ths. If it suddenly appears out of nowhere, there is no way of identifying which is the root, which is the 3rd, which is the 5th and which is the added note. The result is any one of the four notes can be regarded as the leading tone. Some of the notes want to go up a half-step and some of them want to go down a half step but there is no way of telling which is which. The ear is waiting for some other shoe to drop, and it will accept any resolution of the harmonic tension, just to get it over with.
So -- if you go to a G7, then suddenly slide the root (G) up a half-step to G# to form a four note diminished chord (which you can call by any one of four names -- G#dim, Bdim, Ddim, or Fdim -- your choice), you can then resolve it to A, C, Gb, or Eb. But going back to the C would make the exercise kind of fruitless, because you just came from there.
This is why jazz musicians in particular like diminished chords (the four-note kind). You can toss in a diminished chord, modulate to another key, toss in another one and go to yet another key, and on through the night. . . .
The fact that a B, D, and F> are all you need to constitute a diminished chord (as I say, by definition) doesn't change the standard practice of adding a G# to the triad. It just (I hope) clarifies the use of terminology.