I'm confused as to the status of the SSR and SHIELD. Sometimes they seem to indicate that SHIELD was just a new name for the SSR, but other times that it was a completely new organisation. Is there any definitive info available on this in the MCU?
The SSR is the precursor to SHIELD, and once SHIELD was formed, it basically "absorbed" the SSR's resources into their science division. You can see this founding happening in the one-shot "Agent Carter" (not to be confused with the TV show, which shows the SSR pre-SHIELD), and references are made to this throughout Agents of SHIELD as well as a few MCU movies.
During WWII, the Strategic Scientific Reserve, or SSR, was formed by the US to combat the corresponding Nazi special weapons program (HYDRA). Howard Stark originally worked for the SSR, as did Dr. Erskine when he created Captain America. Peggy Carter was also a special agent with the group.
After the war, the US continued to operate the SSR, though Howard Stark seems to have stopped working with them. You can see the SSR in operation in Agent Carter. While the SSR did some investigating of criminal activity, it's primary purpose was to collect, research, and protect scientific advancements that could be a threat to the US.
At some point after WWII, the head of the SSR (Col. Chester Phillips), Howard Stark, and Peggy Carter decided to form a new organization, the Strategic Homeland Intervention, Enforcement and Logistics Division. This new group had a broader goal: to protect the world from all forms of terrorism. This new group took the existing SSR and converted it into the SHIELD Science Division (e.g. this is the group that would eventually hire Armin Zola).