What field(s) of engineering deals specifically with creating/innovating stuff

Examples of stuff to create/innovate are a beard shaver, a cash register, a computer, etc.

That depends. All engineering contains some level of "Design Engineer". However, you would probably like Mechanical Engineering to get the most well-rounded exposure to design processes. As a mechanical engineer we covered, very broadly, thermal cycles in power plants, levels and mechanisms, static loading and dynamic reactions (a trigger releasing a spring loaded mechanism for example), electrical circuits (you will find out in mechanical classes that you need 1hp torque; and an electrical circuit class tells you how much voltage you need to get that from a motor).

Of course, electrical engineering classes design circuits. Civil engineering designs buildings, roads, heating systems, etc. Polymer engineering designs new plastics for different applications. BUT mechanical seemed to be the best for understanding a little bit of everything and really helping to bring it together into a cohesive product.

Creation and innovation is at the core of every engineering field.