This question is inspired by my previous question.
People often write $H^k_{sign}(M;R)$ for the $k$th singular cohomology group with coefficients in $R$. However, I don't understand what this means. To me there are 2 natural definitions. It could be cohomology groups associated to the complex (Where $C_i(M)$ denote the integral singular $i$-chains) $$\text{Hom}(C_1(M),R)\to \text{Hom}(C_2(M),R)\to\dots$$ Or the cohomology groups associated to the complex $$\text{Hom}(C_1(M)\otimes R,R)\to \text{Hom}(C_2(M)\otimes R,R)\to\dots$$ In Hatcher for example, it seems like he mostly uses the second interpretation (see for example the bottom of page 198), but I'm not even sure about that.
To me the second seems more natural; why would we consider homs from integral chains to $R$ when we can consider homs from $R$-chains to $R$.
So my question is, when people write $H^k(M;R)$, exactly which cohomology groups do they mean?