Difference between cross section and probability of interaction

When I read some textbooks I find that the definition of cross section is the
probability. Of incident neutron interact with a matter .but I find many other (cross section of fission, scattering, absorption..)
And I find another definition of probability of interaction in distance $l$ which is given by

$P(l)=1-e^{frac{l}{lambda}}$

Where $lambda=frac{1}{Sigma} $
And $Sigma$ is the macroscopic cross section $Sigma=N sigma$, $N$ is the density of nuclear and $sigma$ is the croos section

My question is what is the difference between the two definition?and if we want to know if an incident particle is interact with nuclear who is the better and give experiment result?

For example if we send neutron into $Po^{209}$ what is all the cross section that we should take it into consideration