You can be a web developer with SEO expertise.
I am a designer, but I have knowledge in SEO which helps me in my job.
From experience those only focused on SEO master short-term black hat techniques with instant results, this is why they need to keep on top of things, as their techniques are mostly frowned upon.
In terms of stability I would have to say web designer. SEO will eventually turn into something else. Just to highlight when I started coding websites as S.E.O. professional was completely different to today's S.E.O. guy. This change will not stop until S.E.O. as you know completely changes into a practice that designers and developers are aware of, rather than a 'secret' 'art form.
I strongly feel we've reached this day, and there is no need for an SEO professional.