On trumpet, certain performers, notably symphonic and brass quintet performers, rely very much on the 3rd valve slide for the intonation of the low C# (B concert) and the low D (C concert). The design flaw of most trumpets finds these two notes extremely sharp.  With the thumb (for the 1st valve slide) the A above the staff (G concert) is adjusted also from it's being "on the sharp side" on most trumpets.  Of the three notes mentioned, the low C# is the sharpest.

Using the traditional grip mentioned above, the ring finger of the left hand can easily manipulate (extend) the third valve slide for the low C# and D to the desired length to compensate for those notes" sharpness without losing a firm grip on the valve casing that would cause embouchure distortion. 

The thumb for the 1st valve slide has a more difficult time, however, without losing some stability in keeping the proper horn angle and firmness of grip. Some performers equip their instrument with "triggers" which are squeezed rather than pushed with the ring finger (for the 3rd valve slide) or the thumb for the 1st valve slide.  This works very well with the traditional grip.

On the other hand (no pun intended), pistol grips and their variations as previously stated are usually used by performers who seldom have any use for a 3rd valve slide, simply because they are either lead players or high note specialists that probably never play in this low register area.  If they have any concern, it is for the A above the staff that would require a 1st valve slide adjustment.  How they work out this problem is either to "lip" the note flat , use false fingering, install a trigger, or find a horn that is ok on the A.

This concludes the first in a series of "Good Fundamentals".