I don't know which sites you're seeing but the ones I use tend to break that out as “partially vaccinated” or “fully vaccinated” vs “fully vaccinated with booster” — which is good since the evidence has shown that there's a big difference between the waning protection against infection (i.e. neutralizing antibodies) versus protection against severe cases (i.e. T cells) even for partial vaccination.

https://ourworldindata.org/covid-vaccinations https://www.washingtonpost.com/graphics/2020/health/covid-va... https://www.nytimes.com/interactive/2021/world/covid-vaccina...

https://covidactnow.org/ uses “at least one dose” for the national map but breaks out their state-level displays as “1+” and “fully vaccinated”.