You should use Custom Post Types. No extra table, no missuse of Settings API.

Use CPTs with Title field, WYSIWYG/textarea (for caption) & a featured image (for the actual slider image). This way later themes can also make use of it. Just name the CPT "slider". 

[Here´s a read why sliders & carousels are not a good choice.](https://ux.stackexchange.com/questions/10312/are-carousels-effective)