Let $M$ be a smooth $m$ dimensional manifold, $\Omega^k(M)$ the $\mathbb R$-vector space of smooth $k$-forms on $M$ and consider the two cochain complexes $$ 0\rightarrow \Omega^0(M)\xrightarrow{\mathrm d}\Omega^1(M)\xrightarrow{\mathrm d}\dots\xrightarrow{\mathrm d}\Omega^m(M)\rightarrow 0,\qquad(1) $$and$$ 0\rightarrow\mathbb R\rightarrow \Omega^0(M)\xrightarrow{\mathrm d}\Omega^1(M)\xrightarrow{\mathrm d}\dots\xrightarrow{\mathrm d}\Omega^m(M)\rightarrow 0\qquad(2). $$I have seen both of these referred to as the de Rham complex of $M$, but in my experience in eg. textbooks (1) is much more common than (2).
Some aesthetic advantages of (1) include the fact that if $N$ is the number of connected components of $M$, then $\dim H^0(M)=N$ for (1) while $\dim H^0(M)=N-1$ for (2) and that the degree $k$ part of (1) always consists of degree $k$ differential forms for $k\in\mathbb Z$ whereas (assuming we want to keep a sane numbering scheme) the degree $-1$ part of (2) consists of constant $0$-forms instead of $-1$-forms (i.e. zeroes, basically).
However (2) has the - in my opinion much more serious - advantage over (1), namely that (2) is locally exact at all degrees, whereas on a nontrivial manifold (1) is never exact at degree $0$.
Indeed, (2) has much more uniform behaviour than (1) and I think it is natural to consider those $0$-forms to be exact that are globally constant functions rather than those that are $0$. For example if $M$ is in a chart domain that is star-shaped with respect to $0$ and $h_1:\Omega^1(M)\rightarrow\Omega^0(M)$ is the standard homotopy operator then for any $f\in\Omega^0(M)$ we have $$ f=f(0)+h_1\mathrm df, $$ which is a homotopy formula at degree $0$ analogous to the usual homotopy formulae $$ \omega=\mathrm dh_k\omega+h_{k+1}\mathrm d\omega,\quad \omega\in\Omega^k(M), $$but this formula fits into the de Rham complex only if $\mathbb R$ is the degree $-1$ part.
A further point which makes (2) the more natural choice is that if $\Omega^k_M$ is the sheaf of smooth $k$-forms on $M$ and $\mathbb R_M$ is the constant sheaf with values in $\mathbb R$, then the sheaf complex $$ 0\rightarrow\mathbb R_M\rightarrow\Omega^0_M\xrightarrow{\mathrm d}\Omega^1_M\xrightarrow{\mathrm d}\dots\xrightarrow{\mathrm d}\Omega^m_M\rightarrow 0\qquad (3) $$is an acyclic resolution of the constant sheaf $\mathbb R_M$ by a sequence of soft sheaves, which allows sheaf cohomology to be applied to the de Rham complex. However it further confounds me that the sequence of global sections of (3) agrees with (2) only if $M$ is connected as the global section space $\mathbb R_M(M)$ consists of all locally constant functions, rather than the constant ones and the sequence of global sections of (3) is in fact always exact at degree $0$.
So I guess my question is, is there any major disadvantage or "gotcha" of the sequence (2) from the point of view of algebraic topology? Or otherwise is there any particular reason why (1) is what is usually called the de Rham complex of $M$ rather than (2)?