I've usually worked in a waterfall method where everything was designed in Photoshop. That means all templates, in 3 variants: desktop, tablet and mobile. So much work for the designers... The client then approved it all and from there the developers took over. There always were a lot of questions on how things should work on mobile, which element should go where etc. This took away a lot of time and energy fir the developers.
In my new job however, I'm both developer and designer and I've decided to design directly in the browser. I haven't designed anything in Photoshop and this way I can show working concepts directly instead of just showing images. :)
To me this is a way more natural working process, because you remove a step and you are working in the environment your design will be used.
Fortunately I work internally, so no clients... Clients often want to sign off things and there's the hard part: you need to convince the client that designing in the browser is faster and better for the end result than designing everything in Photoshop first
To conclude: go with prototyping! ;)
P.s. Just an example of how time consuming it can be: we had a client who's website consisted of 12 templates. The designer fully designed all of them in 3 sizes, so 36 PSDs... Client gave multiple rounds of feedback which meant editing all these 36 PSDs and sending them to the client again for their approval. This went on for 3 rounds... The designer wasn't happy to say the least... :(