It should just apply to the extra flaming damage, since the rune does not say it converts all the weapon damage to fire damage.
When a weapon gains any type of elemental damage, like fire, the entire attack gains the Fire trait. So let's say a +1 Striking Greatsowrd has the Flaming Rune.
If a creature has "Immunities: Fire" or "Resiste Fire 10," would either of those abilities be affected by the Fire trait of the attack, or would they exclusively apply to the 1d6 of extra fire damage added by the Flaming Rune?
The runes have their own damage, they do not turn everything to that particular type. For example, if you have a +2 Striking Weapon, you can have Frost and Corrosive runes in it, which means that your weapon will be dealing its own 2DX of physical damage + 1d6 of Cold +1d6 of Acid damage. All abilities that grant a specific kind of damage (Barbarian instincts, etc) also do not change the type of damage dealt by the weapon, the item or feature must specifically mention that all damage is converted into the specific type, for example the Champion feat Blade of Justice : http://2e.aonprd.com/Feats.aspx?ID=244
"...and you can convert all the physical damage from the attack into good damage."
This is specifically mentioned by the feat. Why? You might ask? Because having a single type can be very useful, and strong, to bypass resistances of certain types of monsters like Incorporeal Creatures that no longer have the 50% damage reduction, they instead have DR Everything (except Force and other specific types of dmg).