Unless you are moving jobs to just do the exact same thing somewhere else (who wants that) you kind of have to take it on yourself to show up the the interview appearing like an expert in skills you hope to get to really grow comfortable with on the job. You may even need to exaggerate your experience.
Ultimately, specialization should be more about how you think than what facts you know, but the hiring market won't see it that way, meaning you need to just get really good at faking it and not be afraid to back it up later after some late-night study sessions.
The other half of this is that 'levels' don't always translate across environments. Just find work that resonates with you, and you will move up. Obsessing over the level (or even the pay) is probably limiting.