Picture in index.html file will load if I include "index.html" in the URL, but not otherwise

Here is the strangest problem I've ever encountered. I have an index.html file as the home page on my personal website. In that file, I have a link to a graphic (jpg) file. I have a duplicate of the site on my personal computer. The link works fine there. It also works fine online if I write "index.html" after the site name (e.g., "www.mywebsite.com/index.html"). But if I don't (www.mywebsite.com), I get everything I expect to get EXCEPT this one particular graphic. It doesn't show up, and the space it normally occupies is gone as well. It's as if the line of code in the index.html document didn't exist. The link, by the way, is to another URL, and if I copy and paste that URL into my browser, I get the graphic with no problem, so there's no issue of it being an invalid link.

I hope I'm explaining this properly. The key issue is that when I input the web address into the browser field without adding "/index.html", I get a different (and wrong) result from when I do input "/index.html". The problem occurs in all browsers I've tried - Firefox, Chrome, and Edge. Aren't browsers all supposed to search for an index.html file by default? Why should writing it out make a difference, and how do I fix this?

Very grateful for any ideas. I'm really scratching my head over this one.