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The Church is built on Christ the living Stone, the Cornerstone. He has the indisputable authority to make "living stones" and "foundation stones" out of those whom He chooses.

Ephesians 2:20
[The Lord's household]"... is built on the foundation of the apostles and prophets, with Christ Jesus Himself as the cornerstone."

1 Peter 2:4-5 "As you come to Him, the living Stone --- rejected by humans but chosen by God and precious to Him --- you also, like living stones, are being built into a spiritual house to be a holy priesthood, offering spiritual sacrifices acceptable to God through Jesus Christ.

Revelation 12:14
And the wall of the city had twelve foundation stones, and on them were the twelve names of the twelve apostles of the Lamb.

This last image from Revelation shows there were twelve large stones forming the basement of the wall of the New Jerusalem, the names of the Apostles were inscribed on these. The whole Old and New Testament Church is represented in the appearance of the city, but the work of the Apostles receives its special recognition. It is on their teaching and witness for Christ that the great spiritual Jerusalem is built.

There is beautiful harmony of thought here between St. Peter, St. Paul and St. John. St. Paul described the Church as built upon the foundation of the Apostles and Prophets, Jesus Christ Himself being the chief corner-stone (Ephesians 2:20). We may compare the same illustration used by our Lord (Matthew 16:18) and afterwards by St. Peter (1 Peter 2:4-6).

So as the whole thing is built upon Christ, you can see from the context why Christ would designate Simon as "Peter" --- Cephas --- a word which means neither the Chief Cornerstone (Christ) not a bit of pebble or gravel, but a building stone (like all of the 12 referenced in Revelation), made particularly characteristic for Simon because "Rock" (Cephas) is bestowed upon him as his given name.

Whether "petra" or "petros" is used is not important by the time of Koine Greek. You do realize we're talking about Koine Greek, right?

There was some Attic (ancient) Greek poetry, a half-millennium before the time of Christ, in which a distinction was made between "rock" on the one hand and "pebble" on the other (Anacreon, c. 570 BC), but that distinction had disappeared from the language by the time Matthew’s Gospel was rendered in Greek.

The distinction is only found in Attic Greek, but the NT was written in Koine Greek, in which both petros and petra simply meant "rock." If Jesus had wanted to call Simon a small stone in Koine Greek, lithos would have been used.

So as regards the NT, it's a distinction without a difference. From the context itself, we can tell it means foundational building materials for the Church and for the New Jerusalem, not a river-gravel footpath.

Any possible doubts are settled by the Biblical Aramaic.

In Paul’s epistles--- eight timesin Galatians in 1 Corinthians---we have the Aramaic form of Simon’s new name preserved for us. Kepha. In our English Bibles, Cephas.

"And what does Kepha mean? It doesn’t mean a little stone or a pebble. It means Rock. What Jesus said to Simon in Matthew 16:18 was this: 'You are Kepha, and on this kepha I will build my Church.'

From the context itself, we know it is applied as an ongoing identity, for Peter (Cephas), because it's a name-change. It's just as significant for Simon --> Peter as for Abram --> Abraham or Jacob --> Israel. It's a defining change. It's how he's referred to in the NT.

I am happy to go this far into the very enlightening subject of language, because the topic here is the Church. And this is the Church of Sts. Peter and Paul.

Which is to say, the Church founded by Christ.