The first two uses came from probability theory, and are somewhat related as terminlogy. They are however, as far as I know, not related at all with the third concept.
In particular, given a measurable map $f:(X,M)\to(\mathbb{R},\mathcal{B})$, and a measure $\mu$ on $(X,M)$. $f$ (or I guess $|f|$ in your case) defines a push-forward measure $f_*\mu$ on $(\mathbb{R},\mathcal{B})$ by the definition that, for every $b\in\mathcal{B}$ the Borel sigma-algebra, $f_*\mu(b) = \mu(f^{-1}(b))$. Then the distribution function $\lambda_f$ is something like $1- F$ for the distribution function corresponding to the pushforward Borel measure $|f|_*\mu$. (The 1 should be replaced by the total mass of the measure $|f|_*\mu$ when it is not a probability measure.)
See the website Earlist Known Uses of some of the Words of Mathematics for some references for what I write below.
Now, the distribution in the sense of the continuous linear functional is introduced by Laurent Schwartz, in French. In French, however, the distribution function of your Borel measure (or of your measurable function) is called "la fonction de répartition", which strongly suggests that Schwartz's choice of terminology is completely independent of the probability/measure theory uses of the words.
In German, the language in which "distribution functions" were introduced, the probabilistic concept is Verteilungsfunktion, while the functional analytic concept is just taken straight from French/English as Distribution.
The above all strongly indicates that while senses 1 and 2 are related, they are disjoint for the 3rd use of the word distribution. In fact, English is one of the (perhaps few) unhappy languages in which they coincide.
(Just to confuse you further, there is also a use of the word distribution in differential geometry, which also means something completely different and disjoint from the three senses you listed above.)