I ran into the following definition:
If $u$ is a distribution on $\mathbb{R}^d$, then $u$ is called homogeneous of order $m$ if $u(\lambda x) = \lambda^m u(x)$, $x\in\mathbb{R}^d$. But $u$ is not necessarily a function on $\mathbb{R}^d$, so how to make sense of the notation $u(x)$?
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$\begingroup$Defining $\phi_\lambda(x)=\phi(\lambda x)$ for smooth $\phi$, the requirement is $$ u(\phi_{\lambda})=\lambda^{-m-d}u(\phi)\quad\forall\phi\in C_0^{\infty}. $$ If $u$ happens to be a continuous function (and hence $u(\phi)=\int u(x)\phi(x)$), this is equivalent to what you wrote.
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