In most languages an informal pronoun is used to address God. In German they use the informal 'du', for example. It's meant to signify a personal relationship with God, as in the way you would talk to your father.
No difference can be detected between the use of ‘believe in’ and ‘believe on’ in the 16th-cent. versions of the Bible, except that the latter was more frequent; it is now archaic.
While some people may believe that there is a difference, there doesn't appear to be one. "Believe on" isn't something I've ever seen or heard, and it appears to be only used because it appears in some translations of the Bible.
As for usage through history, it wasn't just translations of the Bible, it wasn't just Early Modern English, and it wasn't just those two prepositions.
Old English is very different than Modern English; it used the preposition on, with a very different word order:
Þis hé spræc on Iudea-lande: ðær wæs án eowd of ðam mannum þe on God belyfdon on ðam leodscipe.
Middle English had a lot more variety. I suggest checking out the (free) Middle English Dictionary entry. ME still had some of the weird word order (this word order was only used with on, as far as I can see):
Some sources, like this one (from ~1225) even mix in and on:
[...] to bileuen in god.
Ich bileue on þe holie gost.
And, like I said, there were other prepositions that were used, depending on which meaning. From OED (dagger means obsolete):
intr. To have confidence or faith in, and consequently to rely on or trust to, a person or (Theol.) a god or the name of a god.
With in, on, †into, †unto (rare), †of (rare), †upon.
intr. With in, †of (rare), †on, †to (rare). To have confidence in the truth or accuracy of (a statement, doctrine, etc.). In later use also: to have confidence in the genuineness, virtue, value, or efficacy of (a principle, institution, practice, etc.).
There are plenty of examples of these prepositions being used in ME (again, see Middle English Dictionary), but you'll also see a few examples after that. These are from OED's first definition:
They were al content to leue theyr law and to byleue of Iesu chryst. The boke of Duke Huon of Burdeux, c1515
I byleue vpon god & vpon his feyth. Werke for Housholders, 1530
All that should beleeve on him unto eternall life. Israels prayer, 1649
Our Adversaries will not believe of our Holy Apostle, because they think it Idolatrous to pray to a Creature in the very same manner as to the Creator God. A confutation of popery, 1701
To persuade a savage that it is to his advantage to believe on the Lord Jesus Christ. Missionary & Anthropology, 1945
And these are from the other definition:
They do not well beleeve of the primacy of the Bishop of Rome. Christianographie, 1630
Beleeve lesse to your courage then judgement. The history of Polexander in five bookes, 1647
We must be able to believe on the Churches word, before we have read the holy Scripture. J. B. Bossuet's Conf. with Mr. Claude, 1687