In our particular case this line gets blurred. As a small agency that works with small businesses on tight budgets we do both the design and the development of our projects. There is a distinct difference between the two disciplines.
The website design consists of the view and layout as well as the usability blueprint of the site. How and what the site needs to do. When a user clicks on "A" this will happen. How it should act and what it will look like, where certain elements will be places, what images are going to be used, the color theme, the fonts, and all the aesthetics a site has. It is a separate discipline. A website designer need not even know how to code...How to make "it" happen. That is where the website developer comes in.
That said, most professionals that consider themselves website designers know how to code and visa versa for website developers. Usually the differentiation is in ability. There is a reason that there are stand alone website designers. Designers that don't ever touch a code editor and instead use tools like Sketch, Adobe XD, Photoshop, Illustrator and numerous other tools used to exclusively design a website layout. It takes tons of artistic skill and ability to design a usable, functional, effective, and eye pleasing website. Not all website developers have an artistic eye.
Here is a little insider scoop for you. As you can probably guess, there can be quite a lot of friction between the two at agencies that separate the disciplines. There are many times a designer will pass off the layout and the usability specs to the developer, and he/she will take one look at it and be all "ya right", because they have a deeper understanding of what needs to be done to fulfill the designers vision and what their stack and workflow can accomplish. Many times there is a back and forth...change this, omit that, use this, that won't translate on mobile, etc.. but ultimately the developer has to make it happen, and as close to the designers vision as possible.
A lot of website designers use website builders to mask their shallow ability to code. And a lot of web developers use templates and other sites to get ideas about website design...masking their artistic shortcomings. There is nothing wrong with either. Just a reality. Very few people are really really good at both disciplines. That is perfectly understandable considering one is heavily right-brained and the other left.