The two names for the process by which soil and rock are stripped from Earth's surface to expose underlying coal are:

1. Surface mining: This is the most common term and refers to any mining operation that extracts minerals from the Earth's surface.

2. Strip mining: This term is specifically used when referring to the removal of overburden (soil and rock) to access coal seams.

Both terms are used to describe the same process, but "strip mining" is more specific to the extraction of coal.