The electromagnetic form factors of the three-nucleon bound states were calculated in Complete Impulse Approximation in the framework of the Covariant Spectator Theory for the new high-precision two-nucleon interaction models WJC-1 and WJC-2. The calculations use an approximation for the three-nucleon vertex functions with two nucleons off mass shell. The form factors with WJC-2 are close to the ones obtained with the older model W16 and to nonrelativistic potential calculations with lowest-order relativistic corrections, while the form factors with the most precise two-nucleon model WJC-1 exhibit larger differences. These results can be understood when the effect of the different types of pion-nucleon coupling used in the various models is examined.