Is a graph of zero nodes/vertices connected?
These issues are discussed in the classical paper Is the null graph a pointless concept? by Harary and Read. The conclusion is that usually when we think of graphs, we don't think of them as having zero nodes. Definitions which are equivalent for non-empty graphs can differ on the null graph. Whenever we use null graphs we have to be very careful and choose our definitions according to usage.

Given a definition of connectedness, you can check whether the null graph satisfies it. For example, taking your definition (there is a path connecting any two vertices), the null graph is connected since this condition is satisfied vacuously.