지다 becomes 져. Nobody would say 지어 in real life, using "long" versions is a bit literary.
In the same way that for 주다 and 보다 we say 줘 and 봐, instead of the long 주어 and 보아.
내가 졌어 = I lost (nobody would say 지었어)
I don't think we can shorten 짓다 to 져, it seems like it's always 지어, 지었어.
제 한국 친구가 지어 준 이름이에요 = My korean friend made this name for me
져 준 wouldn't sound right, to my ears at least. I'm not a native speaker so I might be wrong.