I often find these designations as quite confusing and most of the companies in my country have various definition about each of the above mentioned post. I was curious what is the requirement from around the world.
Interesting question, this is how I intend them:
Front-end Developer- the main difference can be one word: programming, this person can participate in the design process (visually) and develop mechanisms of action and behavior of elements (during interactions), however the main responsibility is transform design project to working software (implementation process) + along with the connection to the backend.
UX Designer- often companies have different needs depending on the scale, the environment in which they operate and other external factors. I think that UX designer is closer to business and users, can transform business requirements into a specific solution that can be processed into 0 and 1. It can also be a researcher, scientist who collects and analyzes data to improve the existing situation.
UX Engineer- I may be wrong, because I do not have a mental model created for this role, but I think that he knows very well the technical details of the issue with less emphasis on business (as in the case of UX Designer. More technical - technological context than socio-business).