In reading the previous answers, the element that I perceive is missing is that the designer deals with the customer.
In my experience as an instructional designer / developer, the designer is the one that designs the solution to the problem the customer has, and the developer is the one that makes it function the way the designer wants it to.
The designer gathers information from the customer of the web site to determine what the customer is trying to accomplish. Once they know and understand what the customer is trying to do, they design the site in a way that will effectively accomplish that purpose.
Example, if the customer is providing information to new mothers about how to take care of their child (newborn up to age 3), but not sell anything to them, they will choose a color scheme and fonts that appeal to new mothers. The designer will make the site easy to navigate on a mobile device with one hand (because the other hand is holding the baby). And if there are sounds, they will make them soft and baby friendly. If there is music, they will be lullabies or simple tunes for young children. The customer may want to present information specific to the baby's age, so the designer will figure out how to allow the mother to input the baby's birth date (day month, year, or just month and year). They designer can include a place to display a birthday greeting on the baby's birthday. The web site developer, will figure out how to calculate the baby's age and what criteria to add to the database queries to display information that is appropriate to the age of the baby. Also the developer will figure out how to determine if the current date is the baby's birthday and then display the birthday greeting.
On the other hand, if the customer is trying to sell motorcycle accessories to motocross riders, the web designer will choose a color scheme that appeals to tough young men. (Likely with bold colors with lots of contrast.) The sounds will be loud, motorcycle sounds and probably hard rock music. Because the site is trying to drive sales, there will be lots of links that lead directly to sales pages. They will likely include videos of motocross riders. The designer will decide where to put those videos to drive sales higher. They will not be concerned about the viewer using one hand navigation, because they know they will not be on their bike while viewing the web site. To increase sales, the customer may want to have ads for things specific to the season of the year. The designer will choose how to make those ads appear, and make any changes in color scheme that is seasonal. the Developer will do the calculations to determine the season of the year and access the elements of the web page, and the items for sale, specific for that season
Summary, the designer uses their design skills to solve the problem of the customer. The developer makes all the coding happen so that web page functions as it is designed to do.