You’re sort of giving mixed signals, which might be part of the problem of people not knowing how to approach answering your question.
Glossing over the details, web graphics basically take two forms: raster (.jpeg, .png, .gif) and vector (svg). Photoshop has a built in routine for saving and optimizing raster images for web use. Illustrator is perfectly suited for saving to .svg.
There are, however, newer UI/UX (User Interface and User Experience) applications that provides some more specialized and focused environments for producing graphics for electronic interfaces, like websites. Strictly speaking, though, they’re not really necessary — they just provide a combination of tools that can be time-savers for more advanced UI/UX designers and developers.
Despite your considerable experience in more traditional graphic design and creative direction, you really do sound like a novice with website development, so going back to my first reply (I’ve been building website for about 25 years now), I would strongly recommend learning a bit more about the fundamentals of web development.
You don’t need to become an HTML and CSS guru, but without at least a basic conceptual understanding of these things, you’ll find yourself greatly handicapped and running into obstacles. Just one example of many: you’ll absolutely need to understand the concepts underlying responsive layouts in order to create the optimized graphics to best accommodate those layouts. A few quick course on Lynda.com or another high-quality online tutorial site, will be well worth your time in gaining that basic understanding of how things fit together into a bigger picture.
Now, if all you ever intend on doing is picking colors, choosing options and dropping in graphics on a templated online website building application, well, that’s an awfully modest and self-limiting outlook. Even there, though, a little understanding of what’s going on at a deeper level will answer a whole lot of those kinds of questions that, otherwise, will leave you a bit puzzled.
So to more directly answer your specific question about programs, for now, yeah, I’d recommend, at least for the time being, sticking to Photoshop and, perhaps, Illustrator for creating those graphics. If you want to grab a copy of Sketch or InVision or something similar, that’s an option, too, but you’ll find yourself running up against features that will be totally baffling without that more fundamental understanding that I keep coming back to.