Is the new Covid-19 variant beyond the vaccine?

Take a look at how this variant is spreading...

'NEW DELHI ? At Sir Ganga Ram Hospital, a huge facility in the middle of India?s capital, 37 fully vaccinated doctors came down with Covid-19 earlier this month.

The infections left most with mild symptoms, but it added to their growing fears that the virus behind India?s catastrophic second wave is different. They wonder if a more contagious variant that dodges the immune system could be fueling the epidemic inside the world?s hardest-hit nation.".. https://www.nytimes.com/2021/04/28/world/asia/india-covid19-variant.html

"As the country opened up, more transmissible, harder-to-beat variants took hold in India. Sequencing of COVID-19 testing data shows multiple variants, including B.1.1.7 and B.1617, are infecting a growing number of people. Over the last month, new infections have soared across the nation; variants are likely contributing to the surge, though a lack of testing makes their prevalence unclear. Test positivity rates are at 18 percent, and confirmed infection rates could be vastly under-representative, Jameel said."... https://www.pbs.org/newshour/health...st-vaccines-are-made-how-did-that-happen

Questions:

Does the common cold include coronaviruses?

Is the common cold contagious?

Can they distinguish between the common cold and a "variant" of Covid-19?

Have the people who have been getting the "variants" been vaccinated?

Is this going as planned, that vaccinated people are now being exposed to variants/common cold and now their immune system has been disabled and allows similar viruses to bypass it unscathed?