Yes, Fatou formulated the lemma the modern way that Doob refers to. It appears in Fatou's paper Series trigonometriques et series de Taylor, p. 375 (Acta Math., 30 (1906) 335-400), which he presented as his doctoral thesis. It was a side result in proving that if $f$ is bounded and analytic for $|z|<1$, then $\lim_{r\nearrow1} f(re^{it})$ exists almost everywhere. This, in turn, was part of his proof of a general form of Parseval's identity for trigonometric series of summable functions. Fatou's research was personally encouraged and aided by Lebesgue himself.
The details are described in Lebesgue's Theory of Integration: Its Origins and Development by Hawkins, pp. 168-172. Theorem 6.6 in the quote below is what we now call the Fatou's lemma:
"Theorem 6.6 is similar to the theorem of Beppo Levi referred to in
5.3. Fatou proved his theorem as follows: Let $L$ denote a positive number, and consider the set $E$ of all $x$ such that $f(x) < L$. A sequence of functions $g_n$ is constructed by setting $g_n(x) = f_n(x)$ if $f_n(x) <L$ and
$g_n(x)=f(x)$ otherwise. Then for $x$ in $E$, $0<g_n(x)<L$, and $g_n(x)$ converges
to $f(x)$. Lebesgue's Theorem 5.3 applies and shows that $\int_E f=
\lim \int_E g_n <\liminf \int_a^b f_n$. The lemma then follows, since the right-hand side of the inequality is independent of $L$."
The theorem 5.3 referred to is Lebesgue's extension of
the theorems of Arzela and Osgood to convergence of bounded measurable functions. The theorem of Levi was that if $f_n$ is a non-decreasing sequence of summable functions such that $\lim \int f_n$ is finite, then $f(x) = \lim f_n (x)$ is finite almost everywhere and summable with $\int f_n= \lim \int f_n$. It was published by Levi the same year in Sopra l'integrazione delle serie (Milano Ist. Lomb.
Rend., (2) 39 (1906) 775-80) and used by Fubini to prove his theorem on exchanging the order of integration. Here is some broader context from Hawkins:
"Parseval's equality for any summable function, bounded or not, was
obtained shortly thereafter by Pierre Fatou (1878-1929). As his doctoral
thesis Fatou [1906] presented a paper on "Series trigonometriques
et series de Taylor" with the principal aim "to show the advantage that
can be obtained in these questions from the new notions of the measure
of sets and the generalized definite integral" [1906: 335]. He was particularly
encouraged in this undertaking by his "friend H. Lebesgue
who has not ceased to interest himself with my researches and whose
advice has been very useful to me" [1906: 338]."