If the doctor says you have sun damage on your face. Does that mean it is skin cancer?

No. Not at all ...It probably means you need to more careful when you're in the sun to prevent risk for developing skin cancer.
Not necessarily. Sun damage can take the form of dyschromias (pigment changes), wrinkles, actinic keratosis, etc. One could lump skin cancer in with sun damage, but probably not. To a trained eye, it is easy to tell who has spent too much time in the sun without enough protection. Some "sun damage" could be precancerous, so make sure to have appropriate follow up if your doctor recommends it.
No. All it means is it has sun damage. Such skin is at risk of developing cancer in the future.

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