The rules you quoted answer you, with one exception:
No, you cannot make just anything a staff.
However, based on certain named Staves and other Wondrous Items, you could make a quarterstaff, cudgel (or other weapon by GM Fiat) 'staff' that is also an Enhanced Weapon.
From Creating Magic Items (Staves):
To create a magic staff, a character needs a supply of materials, the most obvious being a staff...
And from Staves
Physical Description: A typical staff measures anywhere from 4 feet to 7 feet long and is 2 inches to 3 inches thick, weighing about 5 pounds. Most staves are wood, but an exotic few are bone, metal, or even glass. A staff often has a gem or some device at its tip or is shod in metal at one or both ends. Staves are often decorated with carvings or runes. A typical staff also functions as a walking stick, quarterstaff, or cudgel.
Finally, some examples of weapons with Enhancements that are also Staves:
Staff of Power "..is also a +2 quarterstaff..."
Staff of the Hierophant "...is also a +2/+2 quarterstaff..."