Is there a standard name for this relation property : " aRb --> there is no c different from b such that aRc "?

Such relations are (in my experience) called "functional", in analogy with functions. Indeed, such a relation is a partial function (and actually I've heard "$R$ is a partial function" more frequently than I've heard "$R$ is functional").

Similarly, relations such that for every $a$ there is at least one $b$ with $aRb$ are called "total" (in analogy with partial vs. total functions), or "serial" (although I've heard that one much more rarely). And relations such that for each $b$ there is at most one $a$ with $aRb$ are called "injective" (or "one-to-one") relations.

According to Encyclopedia of Math and nLab, such a relation is called a functional relation on a set.

A functional relation defines a partial function from the set to itself, so you might as well call it a partial function.