As the table below suggests, should you take into account the size of the browser itself when designing web apps? For example, should my artboard size be 1331x548 instead of 1366x768?
You should consider these sizes in the sense that you should think about your design needing to be responsive, but sizing your artboard to an exact pixel value is not at all necessary.
What you should be far more concerned about is what screen sizes will your users be using? If you can determine this by looking at statistics, that's great. Mostly, you need to focus on an approximate 300-1600px width scale, though it should still work okay on screens much larger than that.
When designing a component, think about what it will look like at different screen sizes (phones, tablets, laptops, large screens). If you can do this for all the components you should be fine.
Your website should be responsive to the browser width for the best experience, and images should scale with it. I wouldn't recommend creating images with purposefully specific dimensions considering so many browsers are different, even on desktop. You can (and probably should) achieve this through CSS or some other style formatting. When I created a web application for my A-Level coursework, I used https://www.w3schools.com/css/css_rwd_viewport.asp which was very handy. It also automatically turns the site portrait when narrower than 600px (from my experience).