I don't know if this is of help, but I have seen this idea for defining integration elsewhere, specifically on pages 10-11 of Reed and Simon's Functional Analysis. It would go something like this: let $S$ be the space of step functions obtained as linear combinations of characteristic functions of half-open intervals $[k/2^n, (k+1)/2^n)$. Then $S$ is dense in the space $C$ of bounded piecewise continuous functions continuous to the right, with respect to sup norm. By the two conditions, the definition of $I$ is uniquely determined on $S$ and defines a bounded linear functional on $S$ with respect to sup norm, so it uniquely extends to a bounded linear functional on $C$. This in fact defines the Riemann integral $I$, and it's pretty much exactly what Reed and Simon do, except they don't restrict to dyadic rational endpoints.)
(Then, we can go on and define $L^1[0, 1]$ to be the completion of $C$ with respect to the norm given by $f \mapsto I(|f|)$, and since $I$ is a bounded linear functional also with respect to this norm, it extends uniquely to the completion $L^1[0, 1]$.)
Edit:: Ah, Reed and Simon mention that this approach to the Riemann integral can be found especifically in Dieudonné, Foundations of Modern Analysis, or in Loomis and Sternberg, Advanced Calculus, Addison-Wesley, 1968. Perhaps one can consult these for further history.