UX, is user experience. It can be part of Business Analysis or in recent years there are more jobs asking for UX designers. A user experience in Business Analyst terms would be called a use case, which is basically a story. Listening the user (usually a stakeholder and or sponsor) and designing the software based around these stories. When you define all the use cases (stories) into requirements, they are agreed upon and given a business requirement number. Its encompasses many things but to focus on how the user wants to interact with the software is one example.
Usually in UX design only roles they will ask for UI design, which is User Interface design, so this is using these requirements/use cases/stories to wireframe (prototype) a suitable interface, so the user can interact with the software as outlined in the UX design use cases/business requirements. This requires a different skill set and IMO is the fun part (hence why people wanted to seperate it from BA work lol). You are designing essentially the front end, which is passed on to the Systems Analyst to find the right people for each part, ie the coding behind pressing a button would go to a coder, the artefacts (collection of UI visual ingredients may go to a graphic designer, the I/O from pressing the button might also go to the Database Analyst (DBA), when the button is pressed it might be a search so the DBA needs to design some syntax so the script is accessing the correct tables etc.
In a web development environment is a lot more simple if its not having to access a complex database or use object oriented code. A single person can build a website if they have the right skills, but in large web factories, UX/UI designers do the same thing but they pass the output onto a web developer.
There are other opportunities for UX, for example its used in industrial design. But I wont go any further or it might go over your head.
No UX and UI have nothing to do with public relations outside Customer Relationship Management systems, of which there are a couple of big players.
I think it sounds like you would enjoy Business Analysis and the UX/UI part of it as its asking a lot of questions to get user requirements and deal with internal and external stakeholders, if you those skills you could do well with just some training.