Do the numbers 1 and 2 in E1 and E2 reactions relate to the number of steps in the elimination reaction's mechanism?
No, the numbers do not refer to the number of steps in the mechanism.
They refer to the molecularity of the reaction, that is, to the number of particles that are involved in the rate determining step.
In an E1 elimination, the leaving group departs first to form an intermediate carbocation. Then a base attacks a β hydrogen in the carbocation to form the alkene.
The first step is the rate determining step. Since it involves only the substrate (and not the base), it is called a unimolecular reaction.
The "uni" means "one". That's where the 1 in the name E1 comes from.
Note that an E1 reaction is a two-step reaction.
In an E2 elimination, the leaving group departs at the same time as the base attacks.
There is a single transition state that consists of both the base and the substrate.
Since the rate determining step involves two particles, this is a bimolecular reaction.
The "bi" means "two". That's where the 2 in the name E2 comes from.
Note that an E2 reaction is a one-step reaction.
Here's a video that compares E1 and E2 eliminations.