Solvers
If the "quasi-static conditions" are satisfied, you can use the QS solvers. In that case, the Electro Quasi Static solver would "in principle" give you the correct solution. However, when there are large differences in material properties, the numerical problem can sometimes be too difficult to solve.
This is why Sim4Life has two additional solvers: the Electro Static solver, which neglects ohmic currents, and the Electro Ohmic Quasi-Static solver, which neglects displacement currents. Both make some assumptions on the material properties, but both are much much faster than the Electro Quasi Static solver. They are also more stable. They should be used whenever possible.
[EDIT 09.01.2023] The Electro Quasi Static solver computes both ohmic and displacement currents, and does not make any assumption other than the validity of the quasi-static approximation.
Applications
Electro Static: problems dominated by capacitive effects, e.g. electrodes separated by air or other non-conductors. This is the case for a human body standing next to (but not touching) a high voltage power line, for example.
Electro Ohmic Quasi-Static: problems dominated by ohmic conductive effects, e.g. electrodes connected via good conductors. This is the case for electrodes placed in contact with the human body, for example.
You can find more information on each QS solver in sections 2.6.1 of the documentation (in particular 2.6.1.5 Methods Implemented in Sim4Life).