"first and the second class passengers" was a term current when the song was written, I doubt it was used in the early convict days.

However convict ships also contained officials & marines who could perhaps be considered as first and the second class passengers as they were better treated & housed.

Wikipedia on History of Australia (1788–1850)
Convicts and free-settlers

When the Bellona transport came to anchor in Sydney Cove on 16 January 1793, she brought with her the first immigrant free settlers. They were: Thomas Rose, a farmer from Dorset, his wife and four children; he was allowed a grant of 120 acres; Frederic Meredith, who had formerly been at Sydney with HMS Sirius; Thomas Webb (who had also been formerly at Sydney with the Sirius), his wife, and his nephew, Joseph Webb; Edward Powell, who had formerly been at Sydney with the Lady Juliana, transport, and who married a free woman after his arrival. Thomas Webb and Edward Powell each received a grant of 80 acres; and Joseph Webb and Frederic Meredith received 60 acres each.

The conditions they had come out under were that they should be provided with a free passage, be furnished with agricultural tools and implements by the Government, have two years' provisions, and have grants of land free of expense. They were likewise to have the labour of a certain number of convicts, who were also to be provided with two years' rations and one year's clothing from the public stores. The land assigned to them was some miles to the westward of Sydney, at a place named by the settlers, "Liberty Plains". It is now the area covered mainly by the suburbs of Strathfield and Homebush.

sandra