There are Christians I've known who bristle at the idea that a person can know the exact time they were born again. Maybe not to the minute(!), but perhaps within a day or week or month.
That some Christians can legitimately pinpoint their conversion experience cannot be gainsaid. The apostle Paul is but one example of this phenomenon. Would any Bible scholar say otherwise.? Paul had an epiphany on the Road to Damascus, and his life would never be the same.
So yes, when a person listens to the Spirit of Jesus and believes that he alone can save them from the penalty of sin, they are born again.
This is not to say that a person does not thereafter "grow into" his or her conversion or new birth. Likewise, being born again relatively early in life--as I was--does not mean a person will not have a second (or third or fourth) renewal of their initial belief--as I did.
When I listened to and believed in the Lord Jesus when I was seven, I can say with full assurance that I was born again. Later on in my teens, however (and no, I cannot recall the day or month or even the year!), I told God that if my initial rebirth when I was seven was somehow incomplete, as a teenager I truly wanted in my heart to be a follower of Jesus. At that point, I experienced ana peace that has not faded in over 50 years.
So yes, not only are Jesus's words to Nicodemus in John Chapter 3 reiterated in slightly different wording in John Chapter 5, but there are other biblical expressions that are synonymous, such as the one found in Romans 10:9:
If you declare with your mouth, “Jesus is Lord,” and believe in your heart that God raised him from the dead, you will be saved.