  Recently Koivusalo, Levesley, Ward and Zhang introduced the set of
simultaneously $\Phi$-badly approximable real vectors of $\mathbb{R}^m$ with
respect to an approximation function $\Phi$, and determined its Hausdorff
dimension for the special class of power functions $\Phi(t)=t^{-\tau}$. We
refine this by naturally extending the formula to arbitrary decreasing
functions, in terms of the lower order of $1/\Phi$ at infinity. We also provide
an alternative, rather mild condition on $\Phi$ for this conclusion. Moreover,
our results apply in the general matrix setting, and we establish an according
formula for packing dimension as well. Thereby we also complement a recent
refinement by Bandi and de Saxc\'e on the smaller set of exact approximation
with respect to $\Phi$. Our basic tool is (a uniform variant of) the
variational principle by Das, Fishman, Simmons, Urba\'nski. We also prove some
new lower estimates regarding the set of exact approximation order in the
matrix setting, which are sharp in special instances. For this we combine the
result by Bandi and de Saxc\'e with a method developed by Moshchevitin.
