"Logic Design" vs. "Digital Circuit Design"
To me, the job titles are very similar, but not exactly the same. "Logic design," in my opinion, implies writing VHDL or Verilog to design digital logic to go onto FPGAs, CPLDs, or maybe even ASICs.

"Digital Circuit Design," on the other hand, means to me that in addition to (potentially working on) the HDL stuff, you are also designing the entire digital part of the board/chip/system - including working on the interfaces to the analog stuff. I would also think "digital circuit design" includes less "logic design" and more using ICs and off-the shelf components/IP blocks.

Now, this obviously does not have to be the case. The two job titles could certainly describe the exact same position, that's just my interpretation. If you're looking for a job in that area, the job description can probably provide some more information on what the particular job entails. If it's still not clear, apply to both and then in the interview you can ask for clarification.