The term 'web designer' is rather generic and has no specific definition other than they are involved in some aspect of designing and/or building web sites.
The roles that can be involved in a web design project can vary from project to project, team to team, firm to firm, but here's a list of various roles:
- content strategist
- Information Architect
- Interaction Designer
- User Interface Designer
- Graphic Designer
- Front end developer
- User Experience Designer
- Usability Analyst
- Accessibility Analyst
I'd say any of the above can be lumped into 'web designer'
And then there's roles that I like to call designers as well, though not everyone agrees:
- user/usability testing
- competitive and industry research
- SEO (white hat, not black hat!)
- Business Analyst
- Back End Developers
So, instead of answering your question, I'll spin it around and ask one back of you: What aspects of Web Design do you think you'll like and want to focus on?
Based on your response about wanting to focus on UI and Front End dev, I'd suggest the following:
Front end development is likely an extension of your current skills. You'll want to focus on HTML, CSS and JS. If you know the basics of JS already, you'll likely want to dive into a good framework like jQuery, that will also help with your prototyping.
As for UI design, that's a slightly different background. Most UI designers have a background in art and design. So for that, you'll want to focus on skills such as drawing, diagramming, color theory, form, art, typography, etc. They then combine that with human factor type skills such as usability best practices, heuristic evaluations, interaction design, etc.
In terms of the tools of the trade, well, that's where you're going to have to get used to Photoshop and its kin. No real way around that.