Confusion about 'memoryless' meaning
The definition as you quoted it is only correct for static nonlinearities. The output of a static nonlinearity only depends on the input function directly, and not on its integral or derivatives. Otherwise, the nonlinearity is called dynamic (i.e. having memory). A static nonlinearity can also have memory if it is multi-valued, i.e. if there is more than one possible output value for a given input value (e.g. a system with hysteresis). Finally, a static nonlinearity is indeed memoryless if it is single-valued, i.e. if its output is uniquely defined by its input value:

$$y(t) = F(x(t))$$

where $y=F(x)$ is an injective function (i.e. each $x$ is mapped to only one value $y$).

So in sum you have


dynamic nonlinearity: has memory
static nonlinearity

multi-valued: has memory
single-valued: memoryless