Weird, I just answered almost the identical question, but that person wanted "my God is my judge."
I will give you the modified version of the answer I gave him (dropping the first "my"), which would be:
אלוהים הוא שופטי
"Elohim hu shofti"
The last word, "shofti," is the word shofet (judge) with a first-person possessive suffix. This is a nicer and more formal way to say השופט שלי (haShofet sheli -- literally "the judge that is to me") that you have from your Google-translated answers. I would use it before any of them, but if you wanted to go with the clunkier/less formal version, the first of your translations is the best one.
Since you asked, the difference between each of your answers is that the second one drops הוא, "hu", which translates to the pronoun "he", but in Hebrew is used as a substitute for "is" when describing something with another noun (for "this man is my father," you'd say "this man he my father"). The sentence can still be understood without it, but it's more correct to keep it in.
In the third one, the definite article "ha" (i.e. "the") has been dropped from the word "judge," making it "God is a judge of mine".
And if you're curious why this sounds nothing like the name "Daniel" -- well there are (at least) two Hebrew roots for things involving judgment. The one in "Daniel" is D-Y-N (דין) (e.g. a Jewish religious court is known as a Beth Din, where "Din" means law or justice). The noun here for a person administering justice is "dayan" (דיין), but this typically refers to a person applying Talmudic law, for example to a divorcing couple.
When speaking of a judge as a profession, usually the other root, Sh-P-T (שפט), is used, hence the word "shofet".
If you wanted a Hebrew translation that did, in fact, use the same root as the name Daniel, you could say אלוהים הוא דייני ("Elohim hu dayani"). It sounds a bit weird, but at least it reflects your name. The actual translation of "Dani El" is really "God judged me" more than "God is my judge", hence why it is different. Well, that and compounding it into a name from a sentence.