How to define an independent interference by God?

Types of Unusual Events There are several different categories of uncommon and unusual events.

1. Anomalies are natural events with a pattern
2. Magic includes sleight of hand, which is basically a trick.
3. Satanic Signs include demonic influence.
4. Psychosomatic Events include mental states of mind, such as the power of positive thinking.
5. Providence includes naturally explained phenomenon occurring at opportune moments.

6. Miracles are supernatural acts that by definition defy natural law.

   From the book, I Don't Have Enough Faith to be an Atheist

Providence Of particular interest for this particular question are the latter two. It was likely Providence that someone forgot something at home and turned around just before another car would have hit them. Did God intervene? Perhaps, so. It's not incontrovertible, but is certainly possible.

When I prayed for a job years ago, did God keep me from getting a job at a company that would get bought out and dismantled in few short months? Then did He provide a job at another solid company? Sure, it could be coincidence, so that's not a verifiable "miracle", but it's part of the life of faith and dependence on God.

Miracles

Miracles are really the issue here, though. It must be first stated, though, that if we begin with the mandate that no miracle is evidence unless we are the ones to see it face to face, then that is a bit presumptuous and probably an invalid standard of evidence to demand.

A miracle could certainly be quite verifiable even if we are not there to see it. Several hundred people saw Jesus after the resurrection. The empty tomb, the changed lives of those who saw Him, the explosive growth of the church, the conversion of hardened enemies of Christianity like Paul and others--all of these give substantial evidence to the resurrection. Of course, God's drawing of a person's spirit is only discernible to that person, but it also serves to validate the message.

Then he appeared to more than five hundred brothers at one time, most of whom are still alive, though some have fallen asleep. 1 Corinthians 15:6 ESV

In modern days, there are numerous accounts of very miraculous events occurring, including miraculous healing (medical miracles). I heard a speaker tell of how his wife's body was full of cancer. They prayed diligently that God would intervene somehow, though they knew not what that might mean. Shortly after the surgery began, the doctor came out and sat down with this man and informed him--in shocked tones--that they could find no trace of the cancer that had been all over her body a few short days before. The doctors was not a Christian at that time, but he is now.

The Place of Miracles

It seems that God generally chooses not to give miracles to those who demand to see them before they will believe in Him.

But he answered them, “An evil and adulterous generation seeks for a sign, but no sign will be given to it except the sign of the prophet Jonah. Matthew 12:39 ESV

The sign spoken about here is that Jesus died, was buried, and rose from the dead three days later.

So, why doesn't God show miracles to people? Wouldn't they all believe in Him then? The biblical answers is, "No, they wouldn't believe in Him if they saw miracles."

The Pharisees saw many of the miracles Jesus performed, yet still refused to believe in Him. Jesus rose from the dead, and they still would not believe. They did not deny the miracles, since they themselves were eyewitnesses. Rather, they attributed Jesus' ability to do the miracles to demonic powers (Matthew 9:34).

Some today state emphatically--even as a challenge--that if they themselves saw miracles firsthand, then they would believe. If we can't produce miracles for them, then they claim that proves Christianity is false. However, like the Pharisees, seeing miracles certainly does not automatically result in faith in Jesus, and whether or not Christianity is true certainly does not depend on that either.

Faith and miracles are related, but in exactly the opposite way--Seeing miracles does not produce faith, but faith can result in a person seeing miracles.

Conclusion

So, divine interference can look like what we would expect--truly miraculous events or providential events. The truly miraculous do occur today and can be verified, but miraculous events cannot be demanded of God as proof of Christianity.

Providence includes naturally occurring events at the most opportune times as well as specific answered prayers. As you said, you can certainly give a natural explanation for these, but for the person who is living a life of faith, it's real to them. And that's the whole purpose of God's intervention in their lives--not to prove to the unbelieving, but to demonstrate His faithfulness to His promises to us.

Your question makes the assumption that some good things occur as the result of divine interference, and that others occur by way of chance. This is a faulty assumption.

Every good and perfect gift is from above, coming down from the Father of the heavenly lights, who does not change like shifting shadows. - James 1:17 NIV

So in answer to your question, what would true independent divine interference 'look' like?:

It would look like a good and perfect gift.

Now I know your question isn't asking specifically about "good things," but more generally about determining whether "personal experiences" are divine. I think the same litmus test can be used, though.

Was the personal experience a "good and perfect gift?" If so, it came from God.

Any genuine personal experience with God also won't contradict any other part of scripture.

I believe most protestants (and possibly other groups) value personal experience, but always in light of scripture. Personal experience is considered a very real, and valuable thing, but without an ironclad standard (God's word), it is too easy for it to contradict reality (and even for one experience to contradict a later one, experienced by the same person).