Why is water an Arrhenius acid?
We define Arrhenius acidity with respect to water

That definition is new to me, whatever it is. The definition I know is:

An Arrhenius acid is a substance that dissociates in water to form $\ce{H+}$ ions.
An Arrhenius base is a substance that dissociates in water to form $\ce{OH-}$ ions.

The key concept to answer the question is the self-ionisation of water given in equation $(1)$.
$$\ce{H2O &lt;&lt;=&gt; H+ + OH-}\tag{1}$$
This ionisation happens in water, and the ion product as written in equation $(2)$ is $10^{-14}$.
$$K_\mathrm{w} = [\ce{H+}] [\ce{OH-}] = 10^{-14}\tag{2}$$
Thus, water qualifies as a substance that dissociates in water to form $\ce{H+}$ ions. It also qualifies as a substance that dissociates in water to form $\ce{OH-}$ ions. It is both an Arrhenius acid and an Arrhenius base and thus the only Arrhenius amphoteric compound.