Looking at Wikipedia here, the definition of a scale factor has two definitions of a scale factor which are clearly not equivalent.
Is definition $(1.)$, or comment $(2.)$ the correct one of a "scale factor" in this context? In other words, can the ratio of magnification, dilation factor, scale factor, or similitude ratio be negative?
It states:
$(1.)$ "In Euclidean geometry, a homothety of ratio $\lambda$ multiplies distances between points by $|\lambda|$ and all areas by $\lambda^2$." The next sentence then clearly refers to and defines $|\lambda|$ to be the "scale factor."
However, this is a problem as it later describes $\lambda$ to be the scale factor (i.e. where $\lambda<0$):
$(2.)$ "The image of a point $(x, y)$ after a homothety with center $(a, b)$ and scale factor $\lambda$ is given by $(a + \lambda(x − a), b + \lambda(y − b))$."
I finally found the answer! I appreciate the time others took to help me. Looking on p. 117 (i.e. p.8 of 23 on the pdf) the definition of "ratio of similitude" is given which is the positive real number of a transformation on a metric space of which a dilation is being scaled by. Thus, the first number must be referring to $|\lambda|$ when talking about the ratio of magnification or dilation factor or scale factor or similitude ratio.
Thus, $(2.)$ is wrong and really should say the "ratio" instead of "scale factor."