‘Web designer’ is not a job title that I like as it really does sound like an archaic term and not really even that correct when applied to a person. The definition of the term ‘web designer’ differs depending on who you ask. If you ask an average person on the street a web designer will be exactly what they have experienced when meeting one, no more and no less. If you ask someone who designs online experiences then the definition will be what he/she does on a daily basis, which will probably be different to what another designer does.
When asked what I do as a job I cater the answer to the person that has asked and these range from; Experience Designer (my favourite), Designer, Digital Designer & yes, Web Designer when pushed.
Any design process is just choices and decisions, this colour, that size, this shape, that position, proximity, grouping, interactions etc. There is no mystery here and no tortured artist labouring over the design, there are just fundamental design rules to apply and also less frequently break when a dissonance is required. Everything on the page should have a relationship with everything else.
Make it look good, make it easy to use & understand. That’s what an Experience Designer/Web Designer does.
Oh and sometimes we also just make that button red and move it to the left when the client asks.