Hi Brian,
I think the recommendation about not to use rosin core solder is because it often leaves an unsightly burnt plasticky residue around the joint. Most solders commonly used in electronics applications are rosin cored.
Commonly, solder is a mixture of lead, tin and copper. In electronic applications, you will also find "silver solder" which is a mix of silver, tin and copper (no lead) which has a higher electrical conductivity (less resistance). "Lead Free" solders are commonly tin/copper.
Lead free and silver solders generally have a higher melting temperature than solders containing lead, and I believe you need an iron-clad tip when working with silver solders.
I believe that using a separate flux provides a cleaner joint in that any residue is more easily cleaned, compared with rosin fluxes. Also, the fluxes are somewhat corrosive, the purpose being to clean the joint to allow the solder to flow and adhere to the metals. This is not ideal in an electronicv application, hence the use of rosin cored solders where the flux is "contained" to an extent.