I know that the gold dragonborn in D&D 5e have a 15ft cone as breath weapon, but is that 15ft radius cone or 15ft diameter cone?
15' long, 15' wide at the end
Cones are described in the PHB (p. 204) at the end of Chapter 10 under spellcasting:
A cone extends in a direction you choose from its point of origin. A cone's width at a given point along its length is equal to that point's distance from the point of origin. A cone's area of effect specifies its maximum length.
A cone's point of origin is not included in the cone's area of effect, unless you decide otherwise.
While confusing because this is under spellcasting and the Dragonborn doesn't use a spell to do this, the mechanics of cones/lines/cubes/cylinders/spheres all rely on the mechanics described in that section.
And in this case, Width is the same as Diameter.
A cone has a height (length) equal to its diameter (width)
A vertical line extends from the cone's point of origin (at the bottom). At any point along this line you are a certain distance from the point of origin. That same distance is how "wide" your cone needs to be. For example, going from the point of origin along the vertical line until we reach the first black line requires us to move 10 feet (two squares). Thus the diameter/width of the cone at that point is also 10 feet; this is accomplished by moving five feet in all perpendicular directions. Similarly, if we move until we are four squares away from the point of origin then the cone will have a diameter of four squares, two in each direction.
Thus the 15-foot cone in question would be found by moving three squares away from the point of origin and then moving 1.5 squares in both directions. This is outlined in blue.
Note also that the cone pictured here is two-dimensional while in reality the cone would be three-dimensional. Though using cones fully in three-dimensions gets quite confusing as the line extending from the point of origin need not be parallel to the ground.