Do you have a point, or are you just being pedantic? That is, is there some essence of your position which I haven't adequately addressed? If so, what is that essence?
FWIW Your original statement is referentially transparent wrt the meaning of 'writing' an algorithm ("I specifically referred..."). You were clarifying what you meant by this phrase. So the substitution of your intended meaning into your original statement is justified; in fact, is the entire purpose of your clarification. Otherwise you're just giving a definition with absolutely no context or purpose.