Have you not heard or used this "phrase" before? Here are a few Scriptural uses:
John 1:12
But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name:
John 6:29
Jesus answered and said unto them, This is the work of God, that ye believe on him whom he hath sent.
John 7:39
(But this spake he of the Spirit, which they that believe on him should receive: for the Holy Ghost was not yet given; because that Jesus was not yet glorified.)
John 9:35
Jesus heard that they had cast him out; and when he had found him, he said unto him, Dost thou believe on the Son of God?
John 9:36
He answered and said, Who is he, Lord, that I might believe on him?
John 11:48
If we let him thus alone, all men will believe on him: and the Romans shall come and take away both our place and nation.
John 17:20
Neither pray I for these alone, but for them also which shall believe on me through their word;
Acts 16:31
And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.
Acts 19:4
Then said Paul, John verily baptized with the baptism of repentance, saying unto the people, that they should believe on him which should come after him, that is, on Christ Jesus.
Romans 4:24
But for us also, to whom it shall be imputed, if we believe on him that raised up Jesus our Lord from the dead;
Philippians 1:29
For unto you it is given in the behalf of Christ, not only to believe on him, but also to suffer for his sake;
1 Timothy 1:16
Howbeit for this cause I obtained mercy, that in me first Jesus Christ might shew forth all longsuffering, for a pattern to them which should hereafter believe on him to life everlasting.
1 John 3:23
And this is his commandment, That we should believe on the name of his Son Jesus Christ, and love one another, as he gave us commandment.
1 John 5:13 These things have I written unto you that believe on the name of the Son of God; that ye may know that ye have eternal life, and that ye may believe on the name of the Son of God.
Believing "in" and believing "on" seem to mean the same thing in Scripture. Since doing so is obviously more than simply acknowledging or generally accepting the story of Jesus of Nazareth is true, we can understand that genuine faith is trusting in Jesus to save us. To trust completely our lives and souls to him who died for us is another way of stating it. It means also that our faith is on and in him to deliver us and not in or on the works that we do. Is there a point to this discussion you want to make?
Of course I am familiar with the phrase, but I don't agree that they mean the same thing or that either in modern English necessarily mean what Scripture intended.
I hold that every word of Scripture is as precious as every crumb of the Eucharist and that the exact wording, in the original languages, was perfectly chosen. It was St. Jerome, in reference to his efforts of translating the books determined to be canon into the Vulgate Latin, who said, in so many words, that only the wording of the original language was inerrant and that every other translation and variant of usage injected error into the mind of the reader.
In modern English to believe in means belief that or belief of. (i.e.; do you believe in love at first sight). The phrase "to believe on" is very narrowly used in modern English outside of a theological context, and means to accept implicitly as an object of religious trust or obedience. It comes closest to the Latin "credo", used by the Church in its earliest Creeds, which means complete and total trust. Credo was originally derived from two other Latin words, cor, meaning heart, and do, from the verb meaning to give. In its original meaning, cor-do meant “I give my heart.” The best analogy I have come up with is related to a parachute. If you believe in parachutes you accept that they work. If you believe on a parachute you believe enough to strap one on and jump out of a perfectly good airplane.
However, the original Koine Greek did not use Credo, for those Gospel and NT verses you cited, it used pisteuo. St. John used nearly 100 times in a number of contexts. The only thing I can derive from this is that St. John's use of the word meant to believe on in spite of reasons and pressures to the contrary.
Does the Bishop Budde satisfy this level of belief? That depends on what your definition of Jesus is, doesn't it?
Peace be with you.