Problem: A centurion or others approach Jesus

There's an incident in which a centurion's servant is healed through faith alone. But there's a discrepancy between Matthew and Luke. This is Matthew 8:5-6:

When he entered Capernaum, a centurion came forward to him, appealing to him, "Lord, my servant is lying paralyzed at home, suffering terribly." (ESV)

Now a centurion had a servant who was sick and at the point of death, who was highly valued by him. When the centurion heard about Jesus, he sent to him elders of the Jews, asking him to come and heal his servant. (ESV)

The standard reply doesn't work

At first glance, Matthew and Luke's versions seem to contradict each other. However, many people have noted that "person X did Y" sometimes means "person X got others to do Y". John 19:1 is a classic example. And so, various inerrantists respond that "a centurion came forward to him" just means that the centurion sent messengers. This solution might have worked, but there's another problem with the passages. The story continues with Matthew 8:13:

And to the centurion Jesus said, "Go; let it be done for you as you have believed." And the servant was healed at that very moment. (ESV)

In Matthew's account, when Jesus tells the centurion to "go", this surely means the centurion was actually present. I am grateful to Errancy.com for this point, which I had missed. I have therefore upgraded this problem to Serious.

Other replies don't work

On a different tack, the Apologetics Press suggests that Mark and Luke are actually talking about entirely separate incidents. But that's rather unlikely: in both cases the centurion or his messengers give the same speech about authority. Compare Matthew 8:9 with Luke 7:8.

It is also possible that he [the centurion] came personally to Jesus after he had sent the elders to Jesus.

And Jesus went with them. When he was not far from the house, the centurion sent friends, saying to him, "Lord, do not trouble yourself, for I am not worthy to have you come under my roof. Therefore I did not presume to come to you. But say the word, and let my servant be healed.

[...] And when those who had been sent returned to the house, they found the servant well. (ESV)

It's clear enough that, in Luke's version, the centurion is never present himself.