Actually, there are plenty of interpreted programming languages, for example Perl or Shell Script so that definition is incorrect.

HTML is not a programming language because it only defines form (how things look), and does not control action (executing operations by itself).

The language for Web Development that controls the execution of operations (say: if the user fills a certain field, fetch related data from a server and display it in certain page areas) is called Javascript and is separate from HTML (which existed before Javascript and can exist without it).

Modern Web standards have also moved a lot of the form stuff to yet another language - CSS, Cascading Style Sheets - which is more powerful and reusable, so HTML is more used for the visual structure of the page and less for things like the fonts of the various pieces of text, though it still contains support for that stuff and you can still use it.