Sensible with money: keeping an eye on your spending, be aware where all your money goes.
Mean with money : making the family go without when you don't have to, so for example not letting your child go on essential school trips because you don't want to spend £15 for the day. As opposed to not letting them because that £15 is needed to feed the family.
Sensible : handing on secondhand clothes from one child to the next.
Mean: making the children wear outgrown and damaged clothing rather than replacing it, when you have the means to do so. Again, not when you're actually living on the breadline and genuinely can't afford it.
Sensible with money: doing your research, picking good value holidays, deciding to leave at 5am instead of 8 and spend the saved money on treats once you're there.
Mean with money: never booking a holiday at all because the thought of dipping into the savings you have makes you feel ill.
Sensible: shop wisely, bargain hunt, cook from scratch with cheap ingredients, save meals out for a treat, and enjoy picnics.
Mean: refuse ever to go to a cafe (when you could afford to without causing financial problems), not letting your children meet up with friends anywhere that costs money ever, take the essential kit list from whatever hobby your child has and supply not the things on the list, but the cheaper not-quite-right options and tell them it's character building.
Sensible: save for house renovations, holidays, replacement cars, delay purchase whilst you save for them, and then buy outright rather than go into debt for them.
Mean: have the money available in your eg decorating fund but decide on balance that since no one ever enters your house, your teenage daughter doesn't actually need to change her room from the nursery prints she loved twelve years ago after all.
Critically, for me, the "mean" side of things assumes you can afford to do these things and simply choose not to preferring to see your savings grow faster. They become sensible when you have no money at all. But you said that wasn't the case for you.
It becomes mean I think when your satisfaction at salting away yet more money against a rainy day outweighs your pleasure at eg treating the children to a hot chocolate and a donut after a long cold walk. Or when you find yourselves making do with lukewarm water and no heating for years, because you prefer to watch your savings grow than use them to replace the boiler - fine if it's just you that's affecting, but mean when it means your children are going to school unwashed and with chilblains, and when you could afford to fix it but can't bear to let yourself spend the money.
Sensible: set your children up with savings accounts and encourage them to use them, I still good financial sense into them, including the idea of delayed gratification.
Mean: taking any money they get for birthdays/Christmas/jobs for neighbours, and insisting every penny goes into the savings account and that they must never ever make a withdrawal.