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I want to first start off with this: Cursing, jinxing, and hexing are all perfectly acceptable practices and telling someone else that they will "be smited by the goddess" or that they will "suffer threefold" or "go to hell" for cursing is extensively rude. There is a Very Large chance that they do not follow the same practice as you, so such restrictions which may prevent you from cursing will not prevent them from cursing. However, you can explain your opinion as to why cursing might be a bad option if you aren't going to be rude and judgemental about it.

The reality of cursing, jinxing, and hexing is that these three words do not mean the same thing. They actually have three different meanings, and using them synonymously is using them inaccurately.
A curse: A curse is a spell meant to harm or hinder someone. The idea is usually to cause bad things for someone who has wronged you or a love one. Sometimes as revenge, sometimes as protection, and sometimes you curse based on pure emotion of the moment. Cursing is actually as old as civilization - the Ancient Egyptians are thought to have established one of the first cognative civilizations that truly ran smoothly. Part of their way for protecting the dead and the dead's treasures was by leaving curses on goods, in tombs, and surrounding the family that was left behind. The practice does outdate the pyramids, and outdates the establishment of the cults/priesthoods. Maybe their intent was strange, outlandish, fluffy, or extreme - but cursing is as old as blessing and the two can walk hand-in-hand.
Jinxing: A jinx was originally a spell meant to make something go wrong, but they weren't meant to last. The idea of them was to teach a lesson - whereas a curse was simply meant to exact revenge for wrong-doing. And generally, jinxes were placed on objects of the wrong-doer so that the objects would not work or sell or be made in the way that wrong-doer wished. Done both specifically and as a general spell, jinxes have grown and changed through the times. A thought that evolved and came about in Eastern Europe during the time of the "witch" hunts and the Crusades was that jinxes often required the enlisting of a spirit of some sort - Romans thought them to be done by poltergheists (Latin for noisy ghosts), Germans and Nordics believed them to be done by pixies, etc. As this belief continued past the trials and wars into the industrial revolutions, the idea that gremlins (a technological parallel to pixies) were the beings that carried out jinxes on technology as it began to evolve.
Hexes. A hex is word similar to spell - a hex is not inherently bad or good, just as a spell is not inherently bad or good. A hex was a type of basic magick done under the same circumstances one might do a ritual. Opening a circle, consecrating a space, preparing an altar or shrine, among other practices were involved in hexing. The word hex derives from the Germanic "Hexe" meaning "Witch" or "Hag." Like the word spell, a hex was just a basic performance of magick. It wasn't until the movie industry got a hold of this word that it was used negatively as the sole meaning. Like many other pagan and alternative practices, the media changed the meaning of the word hex for a horror flic.
In conclusion, we see that curse, jinx, and hex do not mean the same thing. While they might have similar ideas involved with the defining, they are not the exact same.

Sources:
Webster's Dictionary.
Britanica Encyclopedia.
Element's Encyclopedia of 5000 Spells.
Christopher Penczak's Gay Witchcraft; Empowering the Tribe.
CNN.com
And several others that are pretty small.