I want my site logo to look like the second image rather than the first. However, when I do the settings in the second image, the site is displayed as "Example - Site" in bookmarks and Google search results rather than "Example Site". Aside from looking bad, this is also terrible for SEO. How can I get the site to look like the second image while displaying the site's title as "Example Site"?
Depends a little bit if you use a SEO plugin (in most of them the appearance of the title (what you see in bookmarks and Google) is just a setting) or which theme you are using. In some of the themes you also have a setting for the title, in other you would have to go into the code.