The question from "nur12" has a potential to be an interview question.
The classification is essentially function based as it has been pointed out. It so happens that most of the towers/colums which does the function for which it is intended, are tall and has skirt and has high L/D ratio.
It is possible to have a horizontal vessel (Although not economically viable)to perform the same functionas a stripper tower/column.
Towers/Column are vertical vessels, where Mass transfer (Physical reaction) takes place. You may have a stripper column, Splitter column.
A vacuum or an atmospheric column is a distillation column. This distillation column may have a combination of Reflux, stripper, splitter etc or there could be separate columns.
These colums/towers typically has trays, packings etc.
When ever there is a chemical reaction we call that equipment (either vertical or horizontal vessel) a reactor.
In a vertical vessel there is neither physical nor chemical reaction. Here the equipment acts as a degasser or a Knock out drum or a scrubber. The separation is provided by an external element such as demister, schontoter (I am not sure about the spelling). Some vessels are used for storage of high pressure fluids or to get condensate or to collect pressurised drain, in such cases you term it as drum.