I found two different formulations of Haar basis in $L^2(\mathbb R)$. For example, in Heil, Christopher. A basis theory primer: expanded edition. Springer Science & Business Media, 2010. we found:
Let $\chi=\chi_{[0,1)}$ be the box function. The function $$ \psi=\chi_{[0,1 / 2)}-\chi_{[1 / 2,1)} $$ is called the Haar wavelet. For integer $n, k \in \mathbf{Z}$, define $$ \psi_{n, k}(t)=2^{n / 2}\psi\left(2^n t-k\right) $$ The Haar system for $L^2(\mathbf{R})$ is $$ \{\chi(t-k)\}_{k \in \mathbf{Z}} \cup\left\{\psi_{n, k}\right\}_{n \geq 0, k \in \mathbf{Z}} $$ Direct calculations show that the Haar system is an orthonormal sequence in $L^2(\mathbf{R})$ and it is also possible to prove that the Haar system is complete in $L^2(\mathbf{R})$.
Instead, in Christensen, Ole. An introduction to frames and Riesz bases. Vol. 7. Boston: Birkhäuser, 2003. we have the following definition:
Given a function $\psi \in L^2(\mathbb{R})$ and $j, k \in \mathbb{Z}$, let $$ \psi_{j, k}(x):=2^{j / 2} \psi\left(2^j x-k\right), x \in \mathbb{R} . $$ If $\left\{\psi_{j, k}\right\}_{j, k \in \mathbb{Z}}$ is an orthonormal basis for $L^2(\mathbb{R})$, the function $\psi$ is called a wavelet. Already in 1910 it was proved by Haar that the functions $\left\{\psi_{j, k}\right\}_{j, k \in \mathbb{Z}}$ constitute an orthonormal basis for $L^2(\mathbb{R})$ for this choice of $\psi$.
In Heil's book we have that the Haar basis is $$ \{\chi(t-k)\}_{k \in \mathbf{Z}} \cup\left\{\psi_{n, k}\right\}_{n \geq 0, k \in \mathbf{Z}} $$ whereas in Christensen's book we have $\left\{\psi_{j, k}\right\}_{j, k \in \mathbb{Z}}$: i.e. the modulation index is $\in \mathbb Z$ and not $\in\mathbb N$ and $\chi(t-k)$ disappears. Why? Are these formulations equivalent?
You have sub-spaces $V_n$, $n\in\Bbb Z$, that are spanned by $\chi_{n,k}$ and the sub-spaces $W_n$ that are spanned by $\psi_{n,k}$.
The projections $f_n$ of any function $f\in L^2$ onto the $V_n$ is a piecewise constant approximation with the detail increasing with raising $n$.
By construction or generally by demand on orthogonal wavelets, $V_1=V_0\oplus W_0$ as orthogonal sum or decomposition. This can now be iteratively extended in both directions, $$ V_n=V_{-m}\oplus W_{-m}\oplus\dots\oplus W_{n-1} $$
The claim of the first statement is now that $V_n$ "converges to $L^2(\Bbb R)$", that is, $f_n\to f$ for $n\to\infty$, while the second statement claims in addition that $V_{-m}$ "reduces to the null-space", or more precisely that for the projections of functions $f_{-m}\to 0$ for $m\to\infty$.
The second claim is thus stronger than the first.