Modern performances often lean toward the former, but there's no question but that Foster and his performers in the nineteenth century would have been familiar with the 'classical' operatic tradition of the day, which was almost exclusively Italian. By that I mean one which involves improvisation, ornaments, appoggiaturas, little cadenzas. Kathryn Cok treated the accompaniments with the freedom of a continuo part. It worked out very nicely. This freedom to improvise is the essence of the baroque era.
Modern performers can be afraid to depart from the written note.
If there is a musician who does do improvisation, it gets a lot of attention, like the wonderful English violinist, Andrew Manze. Some of these were arrangements which were never written down. Dooley and ensemble also transposed the songs down, so that they would lie in a more parlando tessitura. Dooley sees Foster as a master of the perfectly crafted melody, as was Purcell one hundred and fifty years earlier. Foster's melodies are wonderful, influenced by opera and even Schubert songs.