pi: multipliers for dynamics equality constraints
lam: multipliers for inequalities
t: slack variables corresponding to evaluation of all inequalities (at the solution)
sl: slack variables of soft lower inequality constraints
su: slack variables of soft upper inequality constraints

But unfortunately that didnâ€™t make it any clearer.
Where does these variables enter the equations and what can we, as the general user, use these variables for?

I wrote the documentation you quoted and hope to make it more clear.

I think the descriptions for pi and lam can be understood with some optimization knowledge.
Maybe I should write Lagrange multipliers instead of multipliers.

Do you have a question wrt these?

The slack variables t are less intuitive.
They are used internally, to deal with the inequality constraints.
Their initialization has very little influence on the solver.
As a general user, I would not use them.