You (and they) are correct that "Barik" means "send blessings". And they are also correct in saying that "Salli" means "send prayers", but our understanding of it makes the difference.
Is read in Arabic:
Allaahumma salli ‘ala Muhammad wa ‘ala aali Muhammad kama salayta ‘ala aali Ibraaheem, wa baarik ‘ala Muhammad wa ‘ala aali Muhammad kama baarakta ‘ala aali Ibraaheem fi’l-aalameen, innaka hameedun majeed
O Allaah, send prayers upon Muhammad and the family of Muhammad, and send blessings upon Muhammad and the family of Muhammad, as You sent prayers and blessings upon Ibraaheem and the family of Ibraaheem among the nations; You are indeed Worthy of Praise, Full of Glory.
The scholars interpreted the praying to mean:
- It would mean peace — "Allah sends His peace (upon)..."
- It would mean Istighfar (asking forgiveness from Allah for...)
- It would mean making Du'a in their prayers for...
But take into consideration that this specific praying, which is related to the Prophet (Peace Be Upon Him), is more specified than peace or else we would pray for any other Muslim if it always meant peace.
As for the interpretation of:
It means that we are asking Allah to praise him (the prophet Muhammad) in the Upper Skies among the Angles.