My understanding is that quadrivalent flu shots are designed to build immunity to 4 flu virus strains, while trivalent flu shots are designed to build immunity to 3 flu virus strains. Is this accurate?
For years, flu vaccines were designed to protect against three different flu viruses (trivalent). This included an influenza A H1N1 virus, an influenza A H3N2 virus and one B virus. Experts had to choose one B virus, even though there are two very different lineages of B viruses that both circulate during most seasons. This meant the vaccine did not protect against the group of B viruses not included in the vaccine. Adding another B virus to the vaccine aims to give broader protection against circulating flu viruses.
The quadrivalent flu vaccine is also just as safe as the trivalent one.
Studies have shown that vaccines made to protect against four flu viruses have a safety profile similar to seasonal flu vaccines made to protect against three viruses, with similar—mostly mild—side effects
Source for both quotes: CDC page on quadrivalent flu vaccine
Now why would people choose the trivalent one?
For one, the quadrivalent flu vaccine is more expensive. According to the CDC the difference is more than 50 percent. This probably means there is a market for it, so it still gets produced.
Also, some patients (and some doctors) will be cautious about anything they perceive as new and "untested" and thus prefer the old one for a few more years. That's another reason to still produce and ship the trivalent flu vaccine.
If the quadrivalent flu vaccine is not available (for example because not enough was ordered), taking the trivalent flu vaccine is recommended over waiting:
Don’t delay getting a flu vaccine if you cannot locate a quadrivalent vaccine. The important thing is to get vaccinated against influenza.