Are those definitions of universal hash family equivalent?
I'm assuming that the cardinality of Key is bigger than the size of the hashtable m


There are still families with collision probability less than $1/m$, e.g., with three keys and two buckets,

$$\{(2, 1, 1), (1, 2, 1), (1, 1, 2)\}$$

has collision probability $1/3 &lt; 1/2$, and there are other counterexamples for all other parameter settings. The most plausible path to your intuition being true would be if $h(k)$ and $h(l)$ were necessarily independent, but they're not and can in fact be anti-correlated.