I've been thinking about this a lot as I'm on the job search. My current title is "Software Engineer" but I identify as a "designer" and feel my day-to-day work is more along the lines of a "developer" role. Does any of this even matter?
From my experience, developer and software engineer are interchangeable titles, so it would make sense that you'd be doing development work if your title is software engineer.
A designer focuses on the product's overall design from how it functions to how it looks.
The difference between software engineer/developer and designer does matter, as most people and companies you talk to will have wildly different ideas of what you do depending on which you say you are. If you do both, then explicitly say so.
I think that in terms of selling your skill set to an prospective employer the differentiation is mostly up to you. How you communicate your skill set is what matters.
Whatever you title is, you'll want to show that you can think critically about both macro and micro level problems.
For instance a developer or engineer will be able to think broadly about the overall design of the product, how it's used/flow and how the technology of the product can affect the user-experience, and then at a micro level be able to think critically about design patterns, architecture and other engineering concepts.
On the other hand a designer will also need to think broadly about the overall design of the product and how it's used/flow, but they look at it through the lens of user-research. At a micro level I would expect designers to be able to think critically about typography, layout, color, illustration and maybe information architecture and copywriting.
Engineers, designers and developers should ideally have some cross-over skill into each other, as well as be able to understand how their work ladders up to the business (problems/solutions/needs/goals).
Let's try defining the concept "design" this way:
For each part of a business, it's about a design and an implementation.
- Company's design: a business plan. Workers' implementation: A whole product that meets business goals.
- Product manager's design: the market researches and the resulting product design (user stories, navigation logics, etc). The implemention:
Think of this:
[ Business goals, [ Macro product design, [ Micro interaction design, [ visual design, [ Front-end technical architecture, Front-end implementation ]]]]]
A paired  contains a pair of elements, the former is the design and the latter is the implementation. This model is very simplified; in reality, there should be designs that each corresponds to multiple implementation (e.g. apps for multiple platforms)
In this model you find it difficult to define "designer" properly, as long as almost every part can be separated into the design part and the implementation part.
Probably we have to abandon the old school idea of distinguishing designers and engineers, and find a better way to organize the responsibilities within a business.