I’m interested in learning more about jobs that involve designing AI chips. What are the typical job titles to search for to see what skills and responsibilities these jobs have?
The discipline is called Electrical Engineering, and the high-level roles are typically Architects, Design Engineers, and Validation/Verification Engineers. Those roles in turn specialize into e.g. Physical Design Engineer, Pre-Silicon Validation Engineer, RTL Design Engineer, etc. I recommend a book "The Pentium Chronicles" by Robert Colwell, if you want to learn more about this field before diving deeper into technical details.
ASIC, RTL, tapeout, FPGA and Verification are all keywords you can use to find job descriptions. Each company is a tiny bit different, so there is a fair bit of variation.