Channel proteins are located on the peripheral surface of the cell, however, they have the function of integral proteins. Is my understanding correct?
In this case of nomenclature, the context/reference point is the plasma membrane, not the cell - peripheral does NOT mean outside the cell, and integral does NOT mean inside the cell.
Integral proteins are integrated into the membrane, i.e. are embedded in the whole phospholipid bilayer that is the cell membrane, and peripheral means the protein is docked nearby, or associated adjacent to the membrane, and interfacing. They are still "at the membrane" but not integral to it.
By channel proteins, you are likely referring to membrane-spanning (transmembrane) proteins which feature pores that conduct molecules and ions. For example, aquaporins conduct or facilitate the transfer of water molecules across cell membranes, and ion channels such as Na+/K+ exchangers/pumps move cations in and out of the cell across the plasma membrane.
Transmembrane proteins such as (ion) channels or transporters are integral proteins (they are integrated into the membrane, not peripherally associated).
Personal aside: I will say that I don't often hear biologists refer to proteins as integral or peripheral... usually one would simply indicate whether they are transmembrane proteins or not. I guess this may vary by field but this practice is less ambiguous and less likely to throw someone off (as unfortunately was the case with you and plenty of others).