How do We Know Jesus died on the cross?

We know Jesus died on the cross because of the inscription on the cross saying King of the Jews. Christians would not invent this; rather it came from the Jews mocking Jesus. This is called the principle of embarrassment. So we can be very confident Jesus died on the cross by this fact alone. So this is a point of reference believers and non-believers can agree on and move forward with. You have to go to the next stage which is to explain the eyewitness testimony of the Apostles seeing Jesus alive form the dead in various group settings for which we can find no naturalistic explanation.

This principle can also be seen in the baptism of John baptizing Jesus. Christians would not go for this it is presumed, so we can conclude John the Baptist really did baptize Jesus. Then the Holy Spirit comes down upon Jesus. And we don't find Jesus baptizing anyone.

What then are the historical criteria methods?
1. Multiple attestation - of independent sources
2. Dissimilarity - anything that doesn't fit early Christian tendencies yet must be adopted by the early Church and goes against what you would expect, e.g. carry swords to chop of the ear of a servant of a priest in the Garden of Gethsemane. Jesus was not starting a military conquest so His followers should not be carrying swords, and certainly not something Romans would permit. You can crucified for this crime.
3. Social coherence - something as taken as historical like certain rules instituted by Jesus are seemingly out of place because the Church was not formed yet in Jesus' day.
4. Coherence - loosely something that agrees with the above methods but also sheds additional light.