Confusing definition of mol when considering second order rate equations
I think the sentence cited above in the wikipedia article is very confusing.
But the wiki article about the Avogadro constant helped me resolve my confusion, which defines $mol$ via the Avogadro constant:
$$N_A = \left\{N_A\right\}[N_A] = 6.022\times10^{23} \text{mol}^{-1}.$$
Hence
$$k_{on} = k^*_{on}N_A = \left\{k^*_{on}\right\}\left\{N_A\right\}\frac{l}{\frac{1}{[N_A]}s}=\left\{k^*_{on}\right\}\left\{N_A\right\}\frac{l}{mol~s}$$
This definition is also given on the hompege of the National Institute of Standards and Technology (NIST), see here.