We call the mRNA shots "vaccines" because it's what people understand.

Traditional vaccines carry the actual virus itself in small amounts:

A vaccine typically contains an agent that resembles a disease-causing microorganism and is often made from weakened or killed forms of the microbe, its toxins, or one of its surface proteins.

mRNA shots include instructions to make one portion of the virus (the spike protein) that is harmless by itself. After injection, the cells in your arm muscles pick up the mRNA, make the protein, and display it on the cell's surface. Your immune system sees the protein and learns how to make an immune response against it. If you are infected with SARS-CoV-2, your immune system recognizes the same spike protein and can quickly induce an immune response to fight the virus.