The diminished chord takes its name from the chord for which it substitutes.
It is the third of the dominant seventh flat nine chord and when that third is called
the "root" , than the original root of the dominant seventh flat nine is called the generator.

In a diminished seventh, their are four practical and eight possible roots.

As to the first point, an example: The A7b9 chord has C# as the third.
Building the top four notes of the A7b9 will give you C#-E-G-Bb.
This spells out a C# diminished seventh chord with A as the generator.

In practical usage, though, if you see a diminished seventh chord, it is generally
named for the bass motion of the root or if close to a tonic chord it will
either take it's name from the tonic tone or a step higher. Example,
Key of A, A diminished seventh or A# diminished seventh.

If this sounds complicated it's because it is but it can be shown by playing
examples.