The universal appeal of John 3:16 is understandable and many of us Gentile Christians, that is probably one of the first verses we memorized.
Let me propose that when we Gentile Christians read John 3:16, we had the benefit of understanding Paul's Gospel of Grace. Hence when we read about "believes in him", we think of believing in his death, burial and resurrection, like what Paul would say in 1 Cor 15:3-4
But is that what John really had in mind when he wrote about "believe in him"? if you read John 20:31, it said, But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name.
Again, we are back to the Gospel of the Kingdom, which was valid for Jews only. No doubt John's Gospel might have been written after Paul died, and John may have understood what Paul was trying to say, but those are speculations.