How to generalize VC dimension?
If the $VC$-dimension is $d$, then the number of sets that can be cut out from $m$ points by your family is $(em/d)^d$. (This is called the shatter function.)
Every set that you can cut out, can be changed at most $\sum_{i=0}^{tm} \binom mi$ ways to contribute to the $t$-shattering.
So if you want to $t$-shatter a set of size $m$, then you need $2^m\le (em/d)^d \sum_{i=0}^{tm} \binom mi\le(em/d)^d 2^{H(t)m}$ (if $t\le 1/2$) where $H(t)=-t\log_2 t-(1-t)\log_2(1-t)$ is the binary entropy.
From this we get $VC(t)\lessapprox d/(1-H(t))$.
Also note that $VC(t)\ge d/(1-t)$ holds trivially.
Of course, these bounds are general and not only for halfspaces.