If you want to know God, look at Jesus. Jesus is God, the exact radiance, expression and light of God, to the world. One does not have to rely on some mystical induction or revelation. He existed, He lived among us and we witnessed His LIGHT, He died and He rose from the dead. Eye witnesses testified to His miracles and over 500 people saw Him after His death and resurrection and witnessed His ascension.
Why would the Jews of the Old Testament believe in God? Because they experienced His presence, His power, His guidance, His love and forgiveness on a daily basis. He spoke through prophets who gave predictions of future events that played out precisely as they said. The rule was if a so called prophet was misrepresenting God and predicted anything, one thing that did not happen, he was stoned to death. You know the story of Moses. Millions of Israelite slaves were freed from Pharaoh. Why would he do that if he himself did not experience the plagues? So through the generations God blessed them and also punished them. They were the test group, if you will, that was given the LAW and they were the example for the rest of the world, that proved no one could keep the Law. Hence, Jesus was sent to fulfill the LAW for us. Then the gospel spread to the gentiles.
Back to the Spirit. You say there is a weak definition. Well, God made man in His image. What does that mean? It is not physical, it is invisible, which means He gave man will, emotions, a mind and each unique in his own way in that we have different talents, abilities, intelligence - all gifts from God. The mind, emotions and will are what is called the soul. The soul is not exactly the same as the spirit. Animals have souls. So God separates us from the animals by giving us a spirit, a compartment of our soul (highly integrated by the way) which is either dead to God or made alive in Christ. We are born with a dead spirit because of the original sin. It just means we are separated from God. Like a cell phone without a battery, but not quite though, because an unbeliever can still speak to God, they just don't get a clear signal, it's confused, static, distorted or nothing at all. And sometimes they are evil thoughts or ideas as well, which we need to learn how to filter out. Let me say this, that all your thoughts are not self generated. You can get them from reading, listening to someone else, deductive or inductive reasoning as it may be; but also you can receive them from God. He guides us, tells us what we ought to do, go this way, do that and if we don't listen to that first intuitive voice, we screw up. One sometimes argues with this in his own mind, conflicting thoughts, torn in different directions, we resist them or dismiss them. Why? These thoughts and ideas, sometimes new and profound come from outside us, from God or our enemy, Satan.
As you said, we need more than inner revelation. It's true, if people did not get more than intuitions in the past, then it would be understandable why some don't believe. Prophecy is one proof to God's existence, that future events came to be with hair-splitting accuracy. Eye witness accounts of angels, miracles and of course Jesus is evidence. Evidence of His design is all around us. This natural world did not design itself through a mindless series of nature selecting mutations (defects), from slime to Marilyn Monroe - sure.
Back to the soul (mind, emotions, will). There are invisible attributes that are not physical. Your mind, will and emotions are not physical. They are expressed in the physical realm, but when you die, you leave your body and they go with you. As someone said, you can't dissect a brain and see thoughts, ideas, talents, memories. All you see is flesh, blood, chemicals. The mind uses the brain to function in this realm and so a thought is sent by neurotransmitters through the body.
Now the spirit is a special compartment of the soul that is empty when we are born. It needs to be quickened, enlightened, recharged, if you will, by God. And then, something wonderful happens, He takes up residence within our spirit. He prepares you for a divine appointment through a series of events and people that come into your life and His power draws you to Himself. You cannot do this on your own. This is why a Christian can say, I was blind but now I see. He gives us spiritual vision that you CANNOT attain by leaning on your own understanding. Jesus said, "No one comes to the Father unless the Father draws him." Here is a list of fruit of the Spirit which grows in the individual as God prepares you and after you are enlightened, continues. Love, joy, peace, hope, faith, goodness, kindness, patience, self control. These are not physical in nature, they are expressed in many ways outwardly but they are not chemicals that you can take. If that were so, we would have found them and could all give the world then and get along perfectly. No, sin is the problem that keeps us from getting along. It is missing the mark, what doesn't work, a rebelliousness against God and people as well. We all are sinners, we lie, steal, covet, etc.
You don't need to look out there in space for answers. The universe is quite simple compared to this incredible body, soul and spirit that we possess. I always say look at the fetus from conception to nine months later, how it develops. Do the cells communicate with each other, do they have a head foreman that says okay, we have many systems to build simultaneously with intricate organs, fibers, cells all contingent on each other to work ... so if you have any questions, refer to the code. NO, they would be pretty smart cells - smarter than Einstein. This life growth process does not happen by itself. Life is not self generated, or designed by nature, it is created and guided by God