The property (P) indeed characterizes the Sobolev space $W^{1,p}$.
Theorem 1. $f\in W^{1,p}(\mathbb{R}^n)$, $1<p\leq\infty$ if and only if $f\in L^p$ and there is $0\leq g\in L^p$ such that $$ |f(x)-f(y)|\leq |x-y|(g(x)+g(y)) \ \ a.e. $$ Moreover the Sobolev norm is equivalent to $$ \Vert f\Vert_{M^{1,p}}=\Vert f\Vert_p+\inf_g \Vert g\Vert_p, $$ where the infimum is over all functions $g$ satisfying the above condition.
That was proved in the paper:
P. Hajlasz, Sobolev spaces on an arbitrary metric space, Potential Analysis, 5 (1996), 403-415.
Since the characterization does not use the notion of derivative the characterization was used to define Sobolev spaces on metric-measure spaces. By now this is a very well developed part of analysis with plenty of publications.
Regarding characterization (P') this is what I know:
Theorem 2. $f$ belongs to the homogeneous Hardy-Sobolev space $\dot{H}^{1,1}(\mathbb{R}^n)$, if and only if there is $0\leq g\in L^1$ such that $$ |f(x)-f(y)|\leq |x-y|(g(x)+g(y)) \ \ a.e. $$
This result was proved in:
P. Koskela, E. Saksman, Pointwise characterizations of Hardy-Sobolev functions. Math. Res. Lett. 15 (2008), 727-744.
Therefore functions in the Hardy-Sobolev space $\dot{H}^{1,1}(\mathbb{R}^n)$ satisfy (P').
Some comments about relation between the condition (P') and Besov spaces are given at the end.
Moreover, the case $p=1$ is very close to a characterization of the space $W^{1,1}$.
Theorem 3. $f\in W^{1,1}(\mathbb{R}^n)$ if and only if $f\in L^1(\mathbb{R}^n)$ and there is $g\in L^1(\mathbb{R}^n)$ such that $$ |f(x)-f(y)|\leq |x-y|(M_{2|x-y|}g(x)+M_{2|x-y|}g(y)) $$ Where $M_Rg$ is the Hardy-Littewood maximal function with supremum of averages over balls of radii less than $R$.
The proof is much more difficult than that of Theorem 1. Theorem 3 was proved in
P. Hajlasz, A new characterization of the Sobolev space. (Dedicated to Professor Aleksander Pelczynski on the occasion of his 70th birthday.) Studia Math. 159 (2003), 263-275.
For a more elaborate theatment of results related to Theorems 1 and 3, see also:
P. Hajłasz, Sobolev spaces on metric-measure spaces. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 173-218, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003.
There are many papers that study Besov and Triebel-Lizorkin spaces from the perspective of the characterization od $W^{1,p}$ given in Theorem 1, see for example:
P. Koskela, D. Yang, Y. Zhou, Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings. Adv. Math. 226 (2011), 3579–3621.
P. Koskela, D. Yang, Y. Zhou, A characterization of Hajłasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions. J. Funct. Anal. 258 (2010), 2637-2661.
Since $f\in L^{1,\infty}$ belongs (at least locally) to $L^q$ for all $q<1$ these results may apply to your question.