I'm trying to find online what is the difference between disk-integrated vs. disk-resolved, but I can't find a website which does explain them, especially in the context of modelling spectral flux density. Also, since the word disk is being used, it is only applicable to spherical planetary bodies? Is spatially-resolved and disk-resolved the same thing?
The way disk-integrated and disk-resolved is used in your cited paper is nothing special, just indicating optical resolving power in relation to the image of the object of interest:
With a disk-integrated signal they mean a single value as measured over the whole disk (of an image a planet makes - or whatever is imaged as a disk).
When they talk about disk-resolved data, they mean that they have a large enough image of the planet they image so that they can start to discover regional inhomogeneity of the signal, thus make out different regions with different properties.
So the distinction is just that of a spatial resolution: either you can resolve the object of interest and see surface (or cloud) structures (disk-resolved). Or your signal is too faint, your optics too small, and you just get spatially unresolved data, thus disk-integrated over the image of the disk, the whole planet (thus you cannot distinguish anything based on location on the planet - you just measure overall albedo, spectral properties etc)