> or using knowledge I'd gained directly from working for them
I wouldn't be comfortable with that. If you have gained knowledge on Redis, you should be able to use Redis in a personal project. It would be different if you had developed a super secret ranking algorithm that you later re-implemented in another project.
They made it clear (I don't recall the wording) that knowledge I gained for myself while working -- from co-workers, from research done as a necessary part of solving problems, etc -- wasn't the issue; it was specifically knowledge about the business and the topic it covers, which was related to the medical field, rather than knowledge of computer-specific issues. So: software projects relying on my knowledge of XSLT gleaned from working there: fine, not a problem; software projects to prescribe warfarin safely: not so fine. This was a good balance, and I had no trouble with it.
I wouldn't be comfortable with that. If you have gained knowledge on Redis, you should be able to use Redis in a personal project. It would be different if you had developed a super secret ranking algorithm that you later re-implemented in another project.