EMI is a specific term relating to EMC. It describes both unintended radiation of RF energy by a circuit as well as how received RF radiation may impact the performance of a circuit. EMI is tested in many ways, often done in a EMC lab with an Anechoic using specialized EMI receivers and carefully calibrated amplifiers and antennas.
Like Creepage and Clearance, EMI is typically regulated by a Safety specification as well as County specific laws such as those imposed by the FCC in the US, ETSI/CE in Europe, IC in Canada, Etc.
Many times, systems struggle to meet regulatory requirements for EMI. This can often be attributed to use of long wires, PCB traces and sub-optimal system shielding and construction. ASIC integration often greatly reduces EMI in systems because it dramatically shrinks the size of interconnections therefore reducing their effectiveness as an antenna to either unintentionally receive or transmit RF energy.