I have a Yamaha Clavinova CLP-360 with a 1/4" stereo headphone jack; Audio quality is very good through it. I'm thinking about getting a cheap USB audio interface (less than $50, I don't mind used) for recording it to my laptop, but I'm not sure if I need it, and if I could instead use my built-in sound card, a 1/4" adapter, splitter, and 3.5mm to usb audio adapter etc.. I have so many specific questions, but I'll start with: What would be the purpose of an audio interface for recording one-track stereo to my medium to low end laptop? I'm most concerned for quality.
Your Clavinova has a stereo AUX-OUT (next to the AUX-IN, in the back panel, by the power cord). You can connect a stereo output (if your interface accepts 2 channels), or a single mono cable when you connect only the LEFT AUX. Don't use the headphone jack.
As @guidot mentioned, the main benefit of using an interface is that the end result quality is dramatically better. I would never advice to use the analog out from the headphone jack directly into a sound card, as the level is so low that your signal-to-noise ratio (the relative power of the signal, compared to the normal background noise from the connection, RF interference, etc) will make your recordings sounds very poor.
As you mentioned, a used, entry-level interface made anytime in the last 5 or so years is more than enough. I would also recommend a Focusrite Scarlett, as @Your Uncle Bob suggested, which you should be able to find used for around your price point. For hobby usage, most Steinberg, M-Audio, Behringer, Tascam or Line6 models would do.
This depends on entirely, what you intend to with the recording afterwards. If this is just for record/playback the analog recording (even with built-in sound-chip) may do.
For anything more ambitious I would suggest to get a cheap midi interface. The abstraction level is much higher and changing the instrument to something your Clavinova does not provide, surely adds a degree of freedom.