The word "preach" is a generic word originating outside Christianity. As such "preach" does not mean "talk to the lost", but rather means "proclaim something". It can be a proclaiming to people who don't know the information being proclaimed. Or, it can be a proclaiming to people who DO know the information. Or, it can be a proclaiming to people are already part of some group.
Preaching, in a distinctly Christian theological sense, is proclaiming information from the King. It is the delivering of a proclamation or announcement from a King, whether to His subjects or to His enemies. As such it can be delivered to the lost, and it can also be delivered to the church.
For example:
Psalm 40:8-10 KJV
I delight to do thy will, O my God: yea, thy law is within my heart. [9] I have preached righteousness in the great congregation: lo, I have not refrained my lips, O Lord , thou knowest. [10] I have not hid thy righteousness within my heart; I have declared thy faithfulness and thy salvation: I have not concealed thy lovingkindness and thy truth from the great congregation.
This preaching is the proclamation of God's righteousness. As such it is "magnifying God" or proclaiming how great, good, and RIGHT God is. It is done "in the great congregation", a term for God's covenanted, separated, special people. Literally, preaching in the church, preaching in the assembly of God's people.
So, preaching is not strictly limited to the lost, exclusively. Notice also this:
Matthew 26:13 KJV
Verily I say unto you, Wheresoever this gospel shall be preached in the whole world, there shall also this, that this woman hath done, be told for a memorial of her.
Is Jesus commanding an evangelistic sermon text? If so, then every message to the lost must reference this event He spoke of. Or is it not more likely that He is using "preach" in a more generic sense, as meaning as long as the Gospel is being proclaimed, this anecdote concerning this woman's good deed is to be kept in rememberance? That is, the church is to remember her deed, and tell her story, throughout the church's existence in this world?
Acts 8:25 KJV
And they, when they had testified and preached the word of the Lord, returned to Jerusalem, and preached the gospel in many villages of the Samaritans.
The word "testified" is a variant conjugation of one of the words in the original post identified as "preaching" and said to be only for the lost. Yet the context indicates this preaching of the apostles was likely directed to the Samaritan believers with whom the apostles were congregating.
Again, the word preach in the NT generally involves proclaiming good news to those who haven't heard it. But to say preaching is only and always for the lost and never for the church goes too far.