Does a phenotype include the presence of a recessive gene
I agree with one of Alan's comments when he says:

"I think this comes down to whether you include being a carrier of a genetic "condition" a part of your phenotype. It's almost a question of semantics"

To expand on this, I would say there are two major definitions of phenotype, and both are employed by geneticists depending on the context.  Definition one is a bit more naive, and I tend to use it when teaching or trying to explain phenotype to a layman: anything you can observe is a phenotype.  Under this definition, having red hair is a phenotype, not having red hair (or, more directly, having black, brown, blonde, etc. hair) is a phenotype, but carrying the recessive gene for red hair is not a phenotype.  I can observe without checking your genome that you do or don't have red hair, I cannot tell without checking your genome that you have one copy of the recessive gene for red hair.

Definition two is more complete, and I tend to use it more in a lab research setting: anything that is the OUTCOME of a genotype can be defined as a phenotype.  Again, red hair and not red hair (meaning brown, black, blonde, etc) are outcomes defined by the genotype.  However, depending on the context (let's say I'm interested in understanding heritable potential and not the color of hair) I'd say that the phenotypes would be 1) the ability to pass on red hair to progeny (outcomes of genotypes RR, RN, RN in your example), and 2) lack of ability to pass on red hair to progeny (outcome of genotype NN in your example).  In both cases, the phenotype is the outcome of the underlying genes, I just have to make sure I frame my scientific context appropriately.