I see that The Masked Avenger has already got an answer (21h ago at the time of this writing :-). Since I also got my counterexample(s) let me state it here. At least I think that my presentation is perhaps more direct.
Let harp $\ [a;b),\ $ where $\ 0<a<b\ $ are positive reals, be the set of reals $\ t\ $ such that $\ a<t<b,\ $ and $\ \frac ta\ $ is an integer (i.e. $\ a\,|\,t;\ $ the musical name harp stands for homogenous arithmetic progressions).
Let $\ n>2\ $ be an even integer. Consider the following two harps:
$$A\ :=\ [n-1;\, 2\!\cdot\!(n^2-1))$$ $$B\ :=\ [n+1;\, 2\!\cdot\!(n^2-1))$$
Also, let
$$\ C\ :=\ A\cup B$$
Let $\ x:=2,\ $ and
$$\ d_H\ :=\ \frac {\left|\,\{y\in H: x|y\}\,\right|}{\left|H\right|}$$
for arbitrary harp $\ H.\ $ Then:
$$d_A\ :=\ \frac n{2\cdot n+1}$$ $$d_B\ :=\ \frac{n-2}{2\cdot n -3}$$ $$d_C\ :=\ \frac{2\cdot n - 2}{4\cdot n-3}$$
and
$$d_B\ <\ d_A\ <\ d_C$$
for every even integer $\ n>2.$
C O M M E N T S to "version 1" and "version 2"
The text of the Question was modified several times (e.g. in my comments, which later I removed as unnecessary, I provide counterexamples to the very first two versions). Thus let me add some comment which complement the present version (2015-05-07, 00:58 of my Ann Arbor time zone :-).
First of all, and this is extra which was not required by the Question: in my examples the values $\ d_i\ $ belong to the arithmetic progressions of the example. Thus I have satisfied a significantly sharper restriction (which to me was not obvious).
On the other hand, the assumption about values $\ d_i\ $ being prime was new to me. Nevertheless, in my examples it is satisfied each time when my values $\ d_i\ $ which are $\ n-1\ $ and $\ n+1\ $ are twins. Also m $\ x:=2\ $ is prime, and relatively prime to each of $\ d_i,\ $ as well as $\ d_i\ $ and $\ d_j\ $ are relatively prime whenever $\ i\ne j.$
The simplest specific example above is obtained for $\ n:=4.\ $; and then the two harps have $\ 9\ $ and $\ 5\ $ elements.
C O M M E N T S to "Last Version"
We get a counterexample also for the "Last Version"--just let $\ n:=4.\ $ The respective values of $\ M\ N\ $ are $\ M:=3\ $ and $\ N:=24.$