Its OK if you are not using Bosch relays, when I refer to "bosch style" I mean having a relay that has the 85, 86 87, 87a, 30 terminal configuration. A lot of vendors make relays that copy the Bosch relay in this fashion. Most 4 prong relays copy this layout, with the exception of not having the 87a terminal. The fact that your four prong relay doesn't have the terminals labeled leads me to believe that it may have a different configuration than the bosch style.
We need to troubleshoot your wiring setup, starting with the four prong relay. Terminal 85 is normally used for the ground connection and terminal 86 is usually wired to a +12v wire to activate the relay, in your case, it receives 12v from the acc key switch. Before you send 12v to terminal 86, there should be no continuity between the two remaining terminals 30 and 87. So, if 85 is grounded, when terminal 86 receives power you should hear a slight click, indicating that the relay coil contacts have switched, and there should then be continuity between terminals 30 and 87, which carry the heavy current load to the lights.
So unplug all of the relay wiring and lets hook it up and test it as we go. Starting with the four prong relay, ground the terminal you believe to be 85 and hook up the acc key switch power wire to the terminal you think is 86. Have someone turn the key on and see if you hear the slight click of the relay activation or test to see if you have continuity between the other two terminals when the acc key in on. If so, hook the battery feed to what should be terminal 30 and hook the DRL's to the other terminal, which should be 87. On the four prong relay, terminals 30 and 87 are interchangeable, so its not important to really determine which terminal is which. If this is wired correctly, then when you turn on the acc key switch, the DRL's should come on. If not, then the relay is not wired properly and needs to be corrected before proceding.
Assuming the wiring to the four prong relay is functioning properly, then unplug the battery feed to the four prong relay and connect it to terminal 30 on the HID relay and then connect HID relay terminal 87a to the terminal on the four prong relay where the battery feed was. So at this point, the HID relay should only have wires on terminals 30 and 87a. If connected properly, the DRL's should once again come on with the acc key switch. Assuming everything is working as described, then ground HID relay terminal 85 and hook up the OEM low beam switch feed to terminal 86. Now, with the acc key switch on the DRL's should be on. Now, turn on the OEM low beam switch(you should be able to hear the slight click of the HID relay coil switching when the OEM switch is turned on) and the DRL's should go out, if everything is wired properly. If so, turn everything off and hook up the wire to the HID lights to the remaining relay terminal 87 and try the sequence again. Turn on the acc key switch to activate the DRL's, then the OEM low beam switch to see if the lights operate properly.
The 14g wire is marginal for most type of high current lighting. 14g wire at 20 amps is only good for up to a four foot length of wire. I'm not sure how many amps you lights are using, 14g may work, but it will drop the voltage down so that the lights will not be as bright as they could be. Terminals 85 and 86 can be wired with smaller gauge, such as 16-18g wire as they only are used to activate the relay coil, which takes very little current.