On this article and Stein & Weiss (statement 1) and the books by Linares & Ponce and Duoandikoetxea (statement 2), I found the following statements of the Marcinkiewicz interpolation theorem:
If $T:L^{p_0}+L^{p_1}\to L_w^{q_0}+L_w^{q_1}$ where $p_0\neq p_1$ and $q_0\neq q_1$ is a sublinear operator of weak type $(p_0, q_0)$ and weak type $(p_1, q_1)$ then it is of strong type $(p_\theta, q_\theta)$ for the appropriate $p_\theta, q_\theta$.
If $T:L^1+L^r\to L^1_w+L^r_w$ is a sublinear operator of weak type $(1,1)$ and weak type $(r,r)$ then it is of strong type $(p,p)$ for $1<p<r$.
They were stated with the same name, which makes makes me think that they ought to be equivalent by some more or less trivial argument, but I haven't managed to find it. Also, none of the sources I mentioned discuss this. Statement 2 can readily be seen to imply statement 1 if $p_0=q_0<p_1=q_1$ by writing the input as the difference of two positive functions and considering $S(f)=T(f^{1/p_0})^{p_0}$.
So, how can it be shown that statement 2 implies statement 1?
I would guess that it is not easy to prove 1 from 2. However, 2 seems to be the most important special case from 1 and therefore, some authors prefer to give only this special case. If I remember correctly, the proof of 2 is also a little bit easier than the more general case 1.