Bottled cider is called hard cider, but the difference is that it is processed in the same way wine is made. Here’s a clip from Wikipedia under the heading of Cider:
“Steps taken before fermentation might include fruit or juice blending, titratable acidity and pH measurements and sometimes adjustments, and sulfur dioxide and yeast additions.Fermentation is carried out at a temperature of 4–16 °C (39–61 °F). This temperature would be low for most kinds of fermentation, but is beneficial for cider, as it leads to slower fermentation with less loss of delicate aromas. Fermentation can occur due to natural yeasts that are present in the must; alternately, some cider makers add cultivated strains of cider yeast, such as Saccharomyces bayanus.
During the initial stages of fermentation, there are elevated levels of carbon dioxide as the yeasts multiply and begin to break down the sugar into ethanol. In addition to fermentative metabolism of yeast, certain organoleptic compounds are formed that have an effect on the quality of cider, such as other alcohols, esters and other volatile compounds. After fermentation, racking occurs into a clean vessel, trying to leave behind as much yeast as possible. Shortly before the fermentation consumes all the sugar, the liquor is "racked" (siphoned) into new vats. This leaves dead yeast cells and other undesirable material at the bottom of the old vat. At this point, it becomes important to exclude airborne acetic bacteria, so vats are filled completely to exclude air. The fermenting of the remaining available sugar generates a small amount of carbon dioxide that forms a protective layer, reducing air contact. This final fermentation creates a small amount of carbonation. Extra sugar may be added specifically for this purpose. Racking is sometimes repeated if the liquor remains too cloudy.
Apple-based juice may also be combined with fruit to make a fine cider; fruit purées or flavourings can be added, such as grape, cherry, raspberry, or cranberry.
The cider is ready to drink after a three-month fermentation period, although it is more often matured in the vats for up to three years.”
By contrast, fresh cider allowed to sit in the fridge for a week starts to naturally ferment (“Fermentation can occur due to natural yeasts that are present in the must”), gets ‘fizzy’, and will develop an alcohol content of up to maybe 5%. That’s the original source of hard cider. The bottled stuff is essentially Apple wine, given the process used to make it.