The best answer you're going to get is that it's entirely subjective and up to interpretation of the player, writer, composer. As you mention any scale with more than 7 notes/tones will have to have a doubling of a letter/note.
The best way to determine this is it's functionality.
For example using your two examples of the 2 octatonic/diminished scales.
I would name H-W Dim accordingly: 1-b2-b3-b4-b5-n5-6-b7-8
(Here I'm accentuating that this scale has a flat 5, a tritone (diminished 5th) as well as a natural 5 (perfect fifth)
Instead of doubling something less important like the 3rd, I'm making sure the important tones of the scale (diminished 5th and perfect fifth) is shown.
In your second example of a W-H Dim I'd name it accordingly:
1-2-b3-4-b5-b6-bb7-n7-8
Once I again I am choosing to mention the doubly flattened 7th as that is the scale degree which makes it a whole diminished 7th compared to a half diminished 7:
1-b3-b5-b7 = half diminished 7
1-b3-b5-bb7 = whole diminished.
So accentuating the important of the double flat 7 is more important than any other tone /note doubling.