Let $\Delta_p = \{\sum_{i = 0}^p \lambda_ie_i \mid \sum_{i = 0}^p \lambda_i = 1, \lambda_i \geq 0 \}$ denote the usual standard $p$-simplex, with $\{e_i\mid i \in \mathbb{N}\}$ the canonical basis for $R = \bigoplus_{i = 0}^\infty\mathbb{R}$. For each set of vectors $\{v_0,\cdots,v_p\} \in R$, let $[v_0,\cdots,v_p]$ denote the usual map from $\Delta_p$ onto the convex span of those vectors by barycentric coordinates. If $v_i = e_{j_i}$ for all $i$, we call it an affine $p$-simplex.
If $\rho$ is a permutation of the set $\{0,1,\cdots,p\}$, then we can define $$\rho(\Delta_p) = [e_{\rho(0)},\cdots,e_{\rho(1)}]\,.$$ Note that $\{e_{\rho(0)},\cdots,e_{\rho(p)}\}$ is positively or negatively oriented (as a basis of $\mathbb{R}^{p+1}\subset R$) according to the sign of the permutation $\rho$.
For any topological space $X$ and any singular $p$-simplex $\sigma\colon \Delta_p \to X$, we can then define the induced singular $p$-simplex $^\rho\sigma = \sigma\circ \rho$. Let us denote the relation of homological equivalence by $\sigma \sim \sigma'$, i.e., the formal difference $\sigma - \sigma'$ of singular $p$-simplexes is a boundary in $\Delta_p(X)$.
Geometrically, one would expect, for any singular $p$-simplex $\sigma$ and any permutation $\rho$, the following relation on $\Delta_p(X)$: $$\sigma \sim \text{sgn}(\rho){^\rho\sigma}\,.$$ It is simple to verify the claim for $p = 1$, and the claim seems to be false for $p > 1$.
This relation seemed so geometrically natural, that Lefschetz forced it upon his definition of the singular complex of a space (see, for example, his 1933 paper, On singular chains and cycles). It had undesired properties, such as torsion on the chain groups. This and the separate treatment of "degenerate" singular simplexes (those which factor trough an affine $p$-simplex with repeated vertices) were one the reasons that motivated Eilenberg-Steenrod's treatment of singular homology (see Eilenberg 1943 Singular Homology Theory) trough ordered vertices instead of oriented simplexes. One consequence of Eilenberg's definition is that the chain groups are way, way bigger, as no cancellation such as the above are a priori necessary to happen, even modulo boundaries.
Is the contemporary, Eilerberg-esque definition of singular homology equivalent to the one given by Lefschetz, in the sense that they yield the same homology groups $H_i(X)$ for any space $X$?
It seems to me that the latter definition favored computational easiness in exchange for geometrical precision. The choice is easily justified if in the end the resulting homologies are equal, if the loss of such a relation is a mere artifice for making the computations easier. Michael Barr's 1995 article Oriented Singular Homology seems to prove the highlighted claim by saying that the two functors are homotopy equivalences. I haven't understood neither his claim (what does he mean by some complex quotient being an homotopy equivalence? Does it means that they are chain homotopic to the identity?), neither his proof (which is seems to use a lot of machinery from general abstract nonsense). Is an affirmative answer to the question both older and easier to see?