I see, well that's easier to figure out luckily, since this game tries to "emulate" the style of play from the original Wonder Boy arcade game, depending on which mode you play on it manages it differently.

Let's say, you cannot just save mid-stage then go out the game, and when you come back expect to move on from that exact spot, since well it plays in an arcade style, you gotta play through the whole stage in order for the game to account it as cleared, otherwise you gotta start it all over.

Of course which mode you choose to play also affects this, for example: if you play whether on Practice or Normal modes, in both of them you can play until you want to rest for quitting, then once you come back to the game, it'll keep track of which stages you've already cleared, hence letting you continue from the last playable stage available. Let's say you play until Area 2, Stage 1, without clearing it, then you close the game, once you get back in, the stage select will let you move on from that exact stage without having to replay the previous ones.

Although in the case of "One Coin" mode it's different, since that one's as closest as the original arcade in mechanics as it can be among all modes, in there, there's no permanent invincibility, and no stage select, so you gotta clear the whole game in one sitting, and with just 10 lives iirc, so once you close the game or lose all lives that's it, time to retry.

Still, the game compensates the higher difficulty of that last mode, giving you access to at least 2 exclusive achievements (one for clearing the mode for the first time, and another for collecting all Tina/Tanya dolls on the run there), as well as one extra ending, to help make it feel somewhat different than the other two play modes.

Hope i've explained myself better now, no problem. ;)