When calculating SNR, is noise included in signal?
The SNR is the ratio of $S$, the power in the signal $s(t)$, and $N$, the power in the noise $n(t)$. So, $$\text{SNR}_\text{dB} = 10\log_{10}\frac{S}{N}.$$

When using RMS values, we have $S = s_\text{RMS}^2/R$ and $N = n_\text{RMS}^2/R$. The resistances cancel out and then $$\text{SNR}_\text{dB} = 20\log_{10}\frac{s_\text{RMS}}{n_\text{RMS}}.$$