It depends what you want. Yes this is still a sepic converter - primarily because of the coupled inductor and the coupling capacitor and the ability to regulate the output voltage when the input is above or below it. Most sepic converters are used to regulate output voltage in which case the loop stability equations are complicated. Most LED driver sepic converters are used to regulate output current. This is an emergency lighting application, where you need to guarantee operation for a certain lifetime off, say, a battery. therefore you must guarantee a certain INPUT current, not output current. If you are regulating on input current or output current, then the loop equations are different (and possibly simpler) than those of a regulated output design, but you are not discovering anything radical here. You cannot use a circuit that regulates input current to provide a fixed output voltage, so this circuit may be easier to stabilise, but no good for the application required
In each of the architectures above, the regulation loop is different, but the guts of the circuit are essentially the same. The equations governing the main 'power' operation of the circuits are essentially the same (it is a sepic), but, yes, the loop stability equations of each will be different.
By the way, either use individual inductors, or a loosely coupled sepic transformer else the input spikes on the current might make your loop compensation complicated (hence defeating the object of your original post). Incidentally, if you remove the coupling capacitor on a sepic, what do you get? Mr Flyback, the answer might be quite close to home....!