I'm aware that different companies have different definitions for job titles, but in general, is "logic design" the same thing as "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.
At Intel's career center (i.e., jobs.intel) I have found two positions that could help answering this question. According to their website, it is expected from a Logic Design Engineer  to:
Performs logic design, Register Transfer Level RTL coding, and simulation to generate cell libraries, functional units, and subsystems for inclusion in full chip designs. Participates in the development of Architecture and Microarchitecture specifications for the Logic components. Provides IP integration support to SoC customers and represents RTL team.
I have also found a Logic Design Engineer position at ARM  which agrees with Intel's definition. Therefore, in a very short summary, a Logic Design Engineer seems to design, simulate and debug RTL logic.
On the other hand, Intel expects from an Engineer for digital circuit design  to:
Conducts or participates in multidisciplinary research in the design, development, testing and utilization of information processing hardware and/or electrical components, mechanisms, materials, and/or circuitry, processes, packaging, and cabinetry for central processing units CPUs and/or peripheral equipment. Prepares specifications, evaluates vendors, and analyzes test reports. Ensures products conform to standards and specifications. Develops plans and cost estimates and assesses projects to analyze risk. Develops procedures, analysis and design for computer components, products, and systems. Initiates, guides, and coordinates overall design and development of new ideas and products. Responds to customer/client requests or events as they occur. Develops solutions to problems utilizing formal education and judgement.
Those tasks, in my opinion, seem to be more related to "project management" and "decision making" than to the actual implemention (not that this engineer doesn't get its "hands dirty"). Additionally, the Digital Circuit Engineer position seems to comprehend chip fabrication tasks, while the other position doesn't. Accordingly, it seems that for Intel there is a difference between the two job positions. However, I could not found a Digital Circuit Designer position at ARM. This could either mean ARM does not use this definition, or that there is simply no positions open at the moment.
Overall, I would advice looking into which positions each company offers and what they expect from each of them as presented definitions could easily vary for other companies.
Note: References will eventually expire as they are job positions. Therefore, I have inserted all relevant content from them in this answer.