Can an elimination reaction be visualised as the combination of the loss of a leaving group and proton transfer?
Elimination refers to the ends rather than the means. On top of that, it does not necessarily refer to the loss of a proton + a leaving group; it can in general refer to the loss of any small molecule. So, for example, a loss of CO2 is also considered an elimination, even though there is no proton there.

As far as proton + leaving group eliminations go (elimination of HX), the order in which the leaving group leaves and the proton falls off determines what type of elimination it is.

The leaving group can leave first - that is called an E1 elimination.

The proton can fall off first - that is called an E1cb elimination.

And if both of them happen at the same time, it is an E2 elimination.