That's not going to work, unless you are flashing your main beam. The dipped beam goes out when you switch to full beam, so no power to your DRLs.

You want the lights to be on with high beam, and switchable with dipped.
So, go back to your first diagram, with a proper permanent earth or -ve connection for the lights.
Ditch the fuse between the relay and lights. keep the one between the relay and battery.
(fuses should always be as close to the power source as possible, and rated lower than the rating of the wiring used, rather than the device)
The feed for the relay goes to the high beam, thus making the DRLs come on when high beam is on.
To let you turn the lights on without high beam, fit a switch that bypasses the relay.
As mentioned above, these are LEDs, a reasonable quality switch can turn them on and off, no problem.

That switch will let you have the LEDs on anytime, handy for night time repairs and draining the battery.(depending on the power of these LEDS)

If you want to only have the DRLs on when the engine is running feed the switch from an ignition live source. If you can't find one that looks like it can handle the current, use a second relay in parallel with the other, with the switch feeding the relay from the side or tail light.