This paper contributes new insights into discretizing Coulomb collisions in kinetic plasma models. Building on the previous works (Carrillo et al 2020 J. Comput. Phys. X 7 100066; Hirvijoki and Burby 2020 Phys. Plasmas27 082307), I propose deterministic discrete-time energy- and positivity-preserving, entropy-dissipating marker-particle schemes for the standard Landau collision operator and the electrostatic gyrokinetic Landau operator. In the case of the standard Landau operator, the scheme preserves also the discrete-time kinetic momentum. The improvements, the extensions of the structure-preserving discretizations by Carrillo et al (2020 J. Comput. Phys. X 7 100066) and Hirvijoki and Burby (2020 Phys. Plasmas27 082307) to discrete time, are made possible by exploiting the underlying metriplectic structure of the collision operators involved and the so-called discrete-gradient integrators.