I went through all this nomenclature confusion when I started making beverages from my plethora of annual apples.
It didn't make it any easier to learn, when a lot of folks on the winemaking forum to which I belong come from the UK, Europe, AZ and NZ!
As far as I can tell, we Americans are the only folks who call plain apple juice "cider". Here, we must distinguish between "cider" (cold pressed apple juice), and "hard cider", which has the alcohol by volume content around 3%-6%. Our "hard cider" is everyone else's "cider".
More info than you might want:
I make apple juice, apple cider (hard), and apple wine. (We'll mention "Apple Jack" in a bit)
All three start with cold pressed apples after crushing. They should be crushed, not ground, because damaged apple seeds are toxic if they are left in the juice too long.
With apple "juice", the first press is filtered repeatedly until the juice is crystal clear, like you see in the stores. Without preservatives (e.g., potassium sorbate), it has a short shelf life in the 'fridge.
Hard cider is made by fermenting raw apple juice only using the natural sugars in the apples. The yeast of your choice is added to the juice (I've even used bread yeast), and it sits in a bucket for a week or two. You can stop right there, pour off the liquid from the sediment in the bucket and drink, or you can transfer it to a carboy with an air lock and continue until all the sugars have been converted to alcohol. In the latter case, the "hard cider" is back sweetened because all the sweetness in the apple juice was turned to alcohol. There isn't all that much sugar in apples alone, so the maximum is around 6% ABV (alcohol by volume).
Apple wine is the same process as "hard cider", but you had extra sugar so the yeast has more "food" and can make more alcohol. Fermenting apple wine with the extra sugar can result in homemade wine with as high as 18% ABV, while most store wine is around 12%.
Some folks refer to Hard Cider as "Apple Jack", acknowledging it's alcohol content.
"Apple Jack" is actually "freeze distillation". Hard apple cider (or wine) is frozen. The alcohol content has a lower freezing point, so you end up with a core of almost pure alcohol in a block of frozen 'juice'. It's not really something to encourage, because with just a few variables going out of whack, instead of ethanol you can end up with methanol (wood alcohol), and that ain't no good to drink.