Problem with proving that $RP \subseteq NP$ : a non-deterministic TM for a language $L \in RP$
Fix some $L\subseteq RP$. Then:

if $x\in L$ then $P_y[T(x,y)\text{ accepts}]\ge {1\over2}$, therefore exists some $y$ where $T(x,y)$ accepts.
if $x\notin L$ then $P_y[T(x,y)\text{ accepts}]=0$, therefore for every $y$ we have $T(x,y)$ rejects.

Thus, by definition of $NP$, $L\in NP$.
As you can see from this proof, you are correct: $NP$ requires one path while $RP$ requires at least half of the paths