whenever I wash the board with water and soap, the glue becomes visible, as though it is returning to its uncured state
Most PVA-type glues (the majority of woodworking glues used today) are clear/transparent in a thin film but slightly sensitive to water, going cloudy/milky if wetted, which I presume is what you're seeing. It's not much to worry about except aesthetically and the fact that the glued-in patch has remained firm with repeated washing cycles shows that what you've used is likely waterproof enough.
As it does indicate there's a thin smear of glue on/in the wood the surface wood fibres if you want to prevent the problem in future you can lightly plane, scrape or sand the surface. This is in order of preference :-) You'll only need to remove a very small amount to get down to unaffected wood, just a few passes with a smoothing plane set to take a light cut which is equivalent to ~0.008"/0.2mm.
Is this safe for food preparation area?
Only a few glues are actually passed or certified as food safe, however, that doesn't indicate the others aren't safe, only that the maker's didn't bother to pay for the certification. It's likely that most glues can be considered safe for direct food contact once they are fully cured, like virtually all wood finishes.
For some context it's extremely likely that the majority of commercial butcher block countertops and cutting boards are not built using a glue with a food-safe certification because the makers don't consider it important, where invisibly-thin glue lines between boards are all that is coming into contact with the food. And only briefly at that.
What kind of glue should I be using for this?
For working cutting boards, as opposed to mostly decorative boards that see little use (many bread boards and cheese boards) you should ideally be using a waterproof, not water-resistant, glue. So if using a PVA glue you want a Type-I*.
Just to mention, online very often the glue recommended for cutting boards is Titebond III but a little confusingly it's a Type-I PVA.
Foaming polyurethane glues, such as the original Gorilla glue, are also waterproof and although they don't seem to see much use in making boards they are a solid choice.
*A little overview of wood glues here in an early Question on this SE in case it's of help, What is special about wood glue?