Yes, the answers you’re finding are correct. WordPress, like most CMSs, separates the content from the presentation. This means that you put all the templates for styling inside the theme, and then you enter all the content in the editor.
That’s not to say there aren’t other ways to do things, but that’s the most common use case for WordPress and most likely what your client is expecting. Most clients would not have any need for you to convert their HTML pages into PHP ones – that PHP alone wouldn’t bring them much benefit, and if they don’t want to (or can’t) update their existing HTML pages they won’t want to (or won’t be able to) update PHP either. The benefit for them in switching from static HTML to a WordPress site is that they’ll be able to use WordPress’s editor to enter their content, all while knowing it will be shown with the correct header, footer, sidebar, etc. automatically.
So, to complete this task, you’ll want to delve more in to the Theme Hierarchy and it typically makes sense to set up a child theme rather than a theme from scratch for your first WP project. You can probably find a theme that has a layout similar to what your client wants – and then create the child theme to tweak and fine-tune things. Once the theme is set up, someone will also need to input all the content – meaning upload all the images, and create all the Posts and Pages. It sounds like you as the vendor may not have scoped this fully yet to the client’s needs, so if possible, it would be wise to write up a more formal and complete statement of work that specifies who will input the content. Sometimes clients will, other times they expect the vendor to, but setting expectations before you go further will help smooth the process.