I have designed and built a lot of similar, in fact I have also added another resistor in series with the diode. Your snubber circuit is often called an RCD snubber.
If you connect the diode cathode to +Vin then you will clamp the flyback voltage close to zero. Two problems with that:
1) You will similarly be clamping your desired output voltage close to zero, at least if you have a significant load applied. Volts per turn are the same on both primary and secondary windings, neglecting the leakage inductance.
2) The transformer needs volt-seconds to reset. If the volts are near zero then the seconds become very long.
Connecting the diode cathode to +Vin will not work for a flyback transformer!
The circuit you show provides what I might call a two-step waveform with ringing in between. This is caused by transformer leakage inductance which is the whole reason why you need the drain snubber. This is often OK but I was building self-oscillating "ringing choke converters" or "critical conduction converters" where the gate drive waveform was derived from the transformer. I wanted to clean up my gate drive work waveform. By properly engineering my RRCD snubber I can eliminate most of the ringing and get an exponential decay instead. It worked much better in my circuit design, and may be slightly better for EMI.
When driving a relay, there is a lot of winding resistance and you are not trying to transfer power through a transformer. Due to voltage drop in the winding resistance, the relay will reset in a reasonable amount of time...whatever that is. The RCD snubber might be useful with a relay if you wanted to absolutely let the relay open-reset as fast as possible. But relays are mechanical, and doing that to speed a relay might only give you some microseconds of speed, but the mechanical motion of the relay is likely to take milliseconds. So there would not be much benefit.
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