Does anybody know what the specialty is when the aerospace engineer designs the engines for the planes?

Aerospace engineering deals with designing and building machines that fly. It is one of the newest branches of engineering, and began in the 19th century with the first experiments in powered flight. As technology progressed, two specialties emerged; aeronautical engineering, which involves designing aircraft such as powered lighter-than-air craft, gliders, fixed-wing airplanes and jets, autogyros, and helicopters; and astronautical engineering, which focuses on the design and development of spacecraft.

In other words, aeronautical engineers are primarily involved in designing aircraft that fly within Earth's atmosphere, while astronautical engineers work with the science and technology of spacecraft that fly outside Earth's atmosphere, according to the U.S. Bureau of Labor Statistics (BLS).

Getting a mechanical or aerospace engineering degree should be your first step. Then looking around for jobs within the aerospace industry and networking with people in that industry through events, competitions, research, and random opportunities. What a lot of people don't realize is that a lot of employers just want smart people capable of solving their problems. By the time you have graduated with an engineering degree you have solved thousands of problems. Employers want problem solvers and with that you can essentially learn any technical skill. If you want your dream job go above and beyond just the classroom, and network network network. I have yet to get a job by applying for it. Every job I have had was by networking. The reason why I mention this and have gone off in a tangent is because if you want to work on the plane engines, you need to network to get there. Yes you can apply and you might find that specific job which is completely possible and that would be awesome. But knowing people and networking will get you to your dream job, even if that means an aerospace job outside of engines, network and move your way into engines through coworkers and managers. Sorry for the tangent, happy learning!

I have only once heard of a Propulsion Engineer, but I think that was a title that he gave himself. He was both a mechanical and an aeronautical engineer.