I'm wondering if I should mention this when I undergo my review, omit
this detail, or mention it vaguely?
Joe and Kilisi are spot on when it comes to this. Omission is probably the best course of action as the offer is no longer in play.
I don't want to come off as demanding, but at the same time, I generally know my market price and want to convince my company that I'd be happier if my salary were closer to what I think I should be getting.
Getting paid what you're worth is not demanding, but too much of a jump from what you're currently making could be seen as so. This is why negotiation on your base pay is so important as raises are usually expressed as a percentage of that.
I'm curious to know how some of you may have negotiated for a raise upon your annual / semi-annual review
As mentioned previously as well, focusing on the value proposition of you at your company is the best approach. However, I would go a step further and also say to have data to back up your claims of the value you've added. Go over everything you've worked on: projects, process improvements, documentation you've generated, culture fit etc.. Also, I would come up with a plan on how you intend to improve yourself and the company in the coming year. Once again, focus on things that add value to the company and have numbers! Quantification of this value add will have more of an effect than statements alone. An example of this would be "decreased build time by 15%" versus "decreased build time".
If all else fails, another job offer will definitely help with negotiation, but I would be careful with this. Looking for another job is a signal that a person is not happy (even if just from the salary), so staying at a place that you're not happy or you being percieved as not happy has its own pros and cons and is an entire topic by itself.