Truth and Reality are intimately related, but distinct concepts.
Truth is conformity with fact or reality.
For one entity (truth) to conform to another entity (reality), the entities must be, in some measure, distinct from one another.
Reality is the "bedrock" -- it exists whether anyone makes claims about it.
Said another way, reality is that which exists independently of ideas concerning it.
So for example, God existed before humans; sub-atomic particles were a component of reality long before humans knew they existed. These are examples of the presence of objective reality in the absence of truth claims.
Truth exists in the context of a claim about reality. Claims can be true or false as measured against reality. Once someone claimed that sub-atomic particles exist, that claim could be evaluated as true or false, and that claim does not alter actual reality.
Jesus claimed to be "the way, the truth and the life." His claim can be evaluated as true or false via general and special revelation. (To be clear, I believe the evidence clearly suggests His claims are true, and all will one day know whether our beliefs aligned with reality, and subsequently were true -- and therefore represented Truth -- or whether they were false)
So (for clarity) I would not make the claim "I believe in tithing"; I struggle with the many potential meanings of that statement. I might make the claim "The Bible calls us to tithe" -- I can get my head around the truthfulness of that statement. While whether I tithe is related to whether I actually *believe* the claim is true, it also seems to be realted to other factors. I agree that we would be justified in quesioning the veracity of a person's claim to believe they are called to tithe, when they do ot tithe.
So when Jesus said he was born to testify to truth, he came to affirm that we can "really believe" that what God has declared is "really real". He lived a life affirming the reality of this Universe as God created it. And (as an aside) as the author of this universe, God gave us the rules of logic, which allow us to have this discussion!
Truth does not depends on one's belief, in fact just the opposite: beliefs can only be declared to be true if and only if they conform to reality.
So to summarize:
Reality exists independent of any claims against it. Truth exists in the context of claims against reality. Beliefs (which are claims) do not influence Truth; beliefs can be labeled as true or false.
By "testifying to the truth", Jesus affirmed the reality of God's nature as evidenced through general and special revelation.
Confusing "Truth" & "Reality"