When grounding conductors in electrostatic problems, is the reference point of electric potential defined at infinity or the ground?
Ground is a node that is assumed to be at zero potential.  How valid that assumption is depends on what you're using it for.

Earth ground is assumed to be referenced to the Earth.  Electrostatically, that means it can sink or source an arbitrarily large amount of charge without changing its overall charge significantly.  Note that it generally doesn't assume that the Earth has a precisely 1:1 balance of electrons and protons, but it is generally assumed that the overall charge of the Earth is negligible.

A circuit ground is a node in the circuit which is assumed to be zero relative potential for the purposes of defining the potentials on other nodes.  It may or may not be an earth ground.

A ground plane is a node in a circuit which is defined to be at zero potential and is ideally as low impedance as possible.  Since many circuits have more connections to ground than most other nodes, this makes it universally accessible through vias.  More importantly, from the standpoints of SI and EMC, it becomes the default return path for higher speed signals.  It cannot be assumed to have a constant electrostatic charge, as it may be in a device which is portable.

An antenna ground (also sometimes called a ground plane) is an active element of the antenna, and its geometry will affect the performance as much as the antenna will.

Note that the finite speed of light quickly puts the lie to the idea that there exists an ideal ground, since a traveling waveform will mean that the same DC node will have different instantaneous potentials in different locations.  This is usually ignored since the signals relating to it have the same limitation, but only if the geometry is nearly ideal (no significant slots or antennae at the wavelength of interest).

In all, ground is any one of numerous sets of assumptions we make to simplify circuit analysis.  It's important to know which assumptions you're making and what conditions have to exist for those assumptions to be valid.