Definition of tamely ramified

Solution 1:

I think you have yourself answered the most of the questions. But, for the sake of completeness, I would like to write an answer. When $p$ cannot divide $ [l:k]_{insep}$, there could be no inseparable extensions between $l$ and $k$, hence $l/k$ is separable, showing that $(i)$ and $(ii)$ are equivalent. Other implications have already been explicitly answered by you.
P.S. One could show that $l/k$ is separable by the fact that inseparable extensions occur only when the characteristic is $>0$, and when the degree is divisible by the prime characteristic. Hence the conclusion.
Notice that this degree needs not be a power of the prime, as indicated by QiL'8.