Bible changed? Example from Arminianism vs Calvinism

As others have already said, this is a simple matter of changes in meaning in English words over the centuries, and even over decades. Most English dictionaries do not contain too much on the words "should", "would", "shall" or "will", and especially they do not discuss the use of all such words in a single entry. For this kind of thing you need the book recommended by, that great stickler for the use of good English, especially in written communications, Sir Winston Churchill: "H.W. Fowler's (Dictionary of Modern English Usage)". Ideally you need the original Fowler's version and a modern version showing how the language has changed such as "Fowler's Modern English Usage" the latest edition by R.W. Burchfield. In the original version of H.W. Fowler the entry on these four words runs to two pages. Be warned, though, you need to be pretty good at English before you attempt the original.

To be brief the word "should" is used in John 3:16 because the latter phrase is conditional upon the "if" statement (though it is not written as an "if" statement it is essentially an "if" statement) "if you were to believe.... you should not perish" . Modern English is not so fussy about using "should" in a conditional statement and modern versions (very rightly) have gone for the more demonstrative "will" (or "shall").

Your comments on Calvinist v Arminian translations of the Greek is a total "red herring", i.e. a distraction from the true line of enquiry. If you want a fascinating book of phrases like "red herring" you need the wonderfully absorbing dictionary "Brewer's Dictionary of Phrase and Fable":

Red Herring - A deliberate or sometimes unintentional diversion, which distracts from a line of enquiry or a topic under discussion. A red herring (i.e. one dried, smoked and salted) drawn across a fox's path destroys the scent and faults the hounds. It is also known as a Norfolk or a Yarmouth capon".

And, finally, I'm not sure if my discussion here of the meaning of red herring itself qualifies as a red herring.

The Bible speaks of the scriptures being twisted. This is Peter speaking about Paul's letters. Many of which are in the Bible.

16 as also in all his epistles, speaking in them of these things, in which are some things hard to understand, which untaught and unstable people twist to their own destruction, as they do also the rest of the Scriptures. 2 peter 3:16 NKJV

So men do in fact twist the scriptures. This could be in a misinterpretation of the scriptures. Or deliberately changing the words and printing a "wrong" version.

On the opposite end, Christians are encouraged to "rightly divide the word"

Be diligent to present yourself approved to God, a worker who does not need to be ashamed, rightly dividing the word of truth. 2 tim 2:15 NKJV

So while there does exist much twisting of scripture, God has promised us a few different things relating to this issue.

The grass withers, the flower fades, But the word of our God stands forever.” Isa 40:8 NKJV

Jesus says the scriptures cannot be broken

If He called them gods, to whom the word of God came (and the Scripture cannot be broken) John 10:35

Jesus also says

18 For assuredly, I say to you, till heaven and earth pass away, one jot or one tittle will by no means pass from the law till all is fulfilled. Matt 5:18

So we have promises from God that his Word will endure forever, it will cannot be broken, and not even a jot or tittle (tiny letter) will pass away.

So we know that God's word will endure, but also that it will be mishandled and twisted.

The remaining difficulty is to discern between the authentic, and the fraudulent. Jesus gives this promise in John 10

“Most assuredly, I say to you, he who does not enter the sheepfold by the door, but climbs up some other way, the same is a thief and a robber. 2 But he who enters by the door is the shepherd of the sheep. 3 To him the doorkeeper opens, and the sheep hear his voice; and he calls his own sheep by name and leads them out. 4 And when he brings out his own sheep, he goes before them; and the sheep follow him, for they know his voice. 5 Yet they will by no means follow a stranger, but will flee from him, for they do not know the voice of strangers.”

Between God's speaking and our hearing, many steps intervene:

• The prophet hears from God, either face-to-face (Moses), or through dreams, or visions
• The prophet speaks
• The scribe records
• Other scribes copy
• A translator reads and renders the Word in a new language in which the words and idioms do not correspond exactly to their original intent
• That language changes over time
• Translations are revised to accommodate the change in language
• Printers typeset and reproduce the Word
• You hear preachers, family and friends explain the English language (or some other), the Bible and its ideas,
• You read the Bible
• The sum of all you have been taught and experienced and are reading and the Holy Spirit's whispers combine to supply a meaning to you

Theories of Biblical inspiration have something to say about each of these steps. However, no matter which one you subscribe to, it is apparent that at the level of individual words, the Bible is an approximation of God's original message. However, by reading the whole Bible and how the same truths are expressed in different ways throughout its pages, you can narrow down the magnitude of the difference between what God said and what you are hearing. The will's and should's will and should sort themselves out. The Spirit, the Church, godly friends and prayer should clarify things in time.

While Paul Chernoch's excellent answer addresses the general case, in the specific example you cite there is no significant difference in the message. You write " should not or shall not ... which would indicate that it could happen but not guaranteed." But the conditional is only one possible meaning of "should".

Although should can be used to imply the conditional, it is normal to use "should" when referring the the effect of an action.

I picked the stone up off the pathway so that people should not fall over it.

I wrote a note to my boss so that he should know I was off sick.

Neither of the above imply any conditionality - they simply state that the second is the effect of the first. The usage is somewhat formal (and British), and less formal (or British) writers would use "would" instead of "should", but the meaning is exactly the same. The Berean translation clearly has the structure that indicates that meaning is not the conditional

"He sent his son so that people should not perish."

Check out the B2 definition of the word in the Cambridge dictionary.

The same is even more true of 'shall', which does not have any sense that could imply any conditionality at all.

The fact that other translators use a construct that cannot possibly be interpreted as conditional indicates that the original was not a conditional.