Looking at those, I can see straight away that the OE light uses an H4 bulb, (three terminals), whereas the other aftermarket light uses what appears to be a pair of H7s

What you need to know is which is dip, which is main beam, then work from that.

 

it looks like the easiest way is to make a loom using male and female spade connectors.You should have 2 earths and a live now and what you need is 2 of each ,with the live that feeds your bulb carry of 2 lives of one for each beam ,then connect the 2 earths if there on the wrong beam just swap the earths over.The halos connect to your side lights

 

you can cut but its best not to incase you switch back to your old lights anytime.If you can find spades to fit that would work,or some people use there old bulb cut it down and snip the 3 prongs just below the glass and solder the wires to them

 

Don't cut it off, use male spades, then tape the whole lot up.

 

I don't have abugeye, so can't say for sure, but was pretty certain they had two lots of lights in there.
Turn the light over to see what it is.

Sidelight?
Looking at that pic, could it be an electrical leveller?

 

yeh thats a leveller you dont use it.Im about to fit some sti lights and am toying with the idea of giving dip beam a permanent earth in the hope that on high beam you have both beams on together like the sti s .So you might like to try that as im pretty sure wiring the other way its only one or the other at any one time

 

hi i`ve got the bugeye and got the same lights. the problem i had was the halos are just 2 little led`s whitch rust away after 8 months and its very hard trying to replace them.also the amber bulb is a 2 prong and not the normal 1 so thats a nightmaire.but the main problem was they failed an mot because the beam from the lights was just a blob. i got shot an put the standard 1`s back on. not as nice but not worth the hassel.