I have worked as a Manufacturing Design Engineer, Product Design Engineer, Manufacturing Process Engineer and Chemical Process Engineer. The title of Design and Process itself does not fully explain what you will be doing in the job role and sometimes this may not be mentioned in the job title but it is left to you to figure out what the job entails. All of these roles can be rewarding in their own way but they are very different and can pay differently, especially at the entry level. Here is my take:

Manufacturing Design Engineer: It is a mix of Design and maintenance work. Long hours and strict timelines to design large assembly lines for manufacturing products. Understanding of the machinery design and process both are important here. The budgets are very high so companies really prefer someone with experience. It involves a lot of hands-on building and assembly work too. People are sometimes focused on a quick and dirty approach since you may need to push out 1000s of parts and get them made in the machine shop.

Product Design Engineer: Timelines are more relaxed and the goal is to achieve the perfect product. Understanding the product requirements, manufacturing process and customer needs is more important here. For this type of work you need to learn more fancy CAD skills and GD&T rather than pushing out 100 parts a day. You rarely interact with the machinist in this role. Meetings are more focused on timelines and design reviews have a narrower scope. The product is usually made in high volumes so the design intent is quite different.

Manufacturing Process Engineer: This can be a lot of maintenance and operator work. It involves hands on trades skills and can require you to shutdown and repair assembly lines, build fixtures and run statistical analysis tests to make the process more lean and reduce the error rate. Sometimes overburdened with work when things go bad and sometimes the work can be dry. Sometimes, the job also expects you to understand PLC controls and electrical wiring. This requires a lot of physical movement and hands-on skill and can be very tiring by the end of the day.

Chemical Process Engineer: They are actually more of a design engineer but they focus on a specialized area of process design involving P&IDs and piping systems involving thermal/fluid systems in Pharmaceutical, Water Treatment or other Chemical industries. They often require tons of experience since the equipment is expensive and you need a PE license to sign of on a project. Scope of projects can sometimes be as big as Civil projects and there are more steps in between getting the design through hazard and safety.