Your understanding of an integrated amp is correct. Unfortunatly you also need to understand that the output of a phono cartridge is a special circumstance. A MM phono cartridge's output is around 3 to 5 millivolts (MC cartridges are much less). Other source componenets such as a CD player, tuner, reel to reel tape player ect. output is in the 1 to 2 volt range ("line level" voltage). A phono stage is required to boost the millivolt signal and apply the required RIAA equilization to the cartridge signal. The phono stage output is in the 1+ volt range that the integrated amp can now work with. There are some integrateds that have "built in" phono stages but the "line stage" only are the more common format today.
Rick
Rick