First, it is indeed a diminished triad. For triads, there is no such thing as a fully or half-diminished chord. The latter two chords are seventh chords. A fully diminished seventh chord has a diminished seventh (i.e. for Bdim7 that's an A flat: Ab), and a half-diminished seventh chord has a minor seventh, i.e. a Bm7(b5) has an A. Both seventh chords add the seventh on top of the diminished triad B-D-F.
Now for the fingering. First of all, a diminished triad is not used and played very often. Second, there are no good sounding and comfortable six-string voicings. You would normally use 3-string voicings and mute the other strings. This is a technique which is very useful to learn anyway, and which can and should be used for other chord types as well, especially on the electric and steel-string acoustic guitar. For these guitars, thumb muting is a good option, at least for the low E string. Now for some common voicings of a Bdim triad (on d, g, and b; from low to high):
x x 3 4 3 x
x x 9 7 6 x
x x 12 10 12 x
Your thumb can mute the low E string, the finger fretting the D string can mute the low A string, and the high string can be muted with your first finger. This, in addition with some right hand control, should result in comfortable and clean sounding chords. I guess from these 3 voicings it's easy enough for you to figure out the shapes on other combinations of 3 strings.
If you want the root on the E or A string, just drop the root of the first two voicings above by one octave. This will give you
x 2 3 x 3 x
7 x x 7 6 x
The finger on one of the two low strings will also help mute the other low string. In the first voicing above you could double the root on the G string:
x 2 3 4 3 x
In the second voicing above you could add the root on the top string:
7 x x 7 6 7
Note that things get a bit easier if you play fully or half-diminished seventh chords, simply because there's a fourth note which you don't need to mute.
