What is the difference between Linear Interpolation factor and Sampling rate conversion factor?
I'll move my comments to an answer:

With sample rate conversion factor $S$, you want to compute samples at "time" $\frac{i}{S}$ for $i=0,1,2,\dots$. "Time" relating to the index of the previous series. The linear interpolation factor is simply then the fractional part of $\frac{i}{S}$ for each $i$ with the integer part telling which sample index to use (together the next) for the linear interpolation.

Simple example to clarify: upsample a sequence a factor $\frac{10}{9}$. You would like to determine the samples at "time" $0, 0.9, 1.8, 2.7, 3.6, \dots$.  First, start with sample $0$, then, compute the linear interpolation between sample $0$ and $1$ with $\eta = 0.9$. Next sample should be at "time" $1.8$, so $\eta = 0.8$ for samples $1$ and $2$. "Time" $2.7$ gives $\eta = 0.7$ for samples $2$ and $3$ and so on. 

Note that you conceptually can compute the "time" $1.8$ sample from sample $0$ and $1$ with $\eta = 1.8$, but using sample $1$ and $2$ with $\eta = 0.8$ is better. So while $\eta$ does not really have to be $0$ and $1$ the results will be better and there is no advantage to not have it in that range.