The percentage of the population needed to attain herd immunity depends very much on how contagious the pathogen is. Revising upward is not a sign of lying, but a response to the more contagious variants that have arisen (separately) in Britain and South Africa. On the other hand, the percentage of the population who are immune will, at any given time, be larger than the precentage vaccinated. Right now about 5.6% of the U.S. population has had an identified case of COVID-19 (and just shy of 1% of the population has died). Estimates of the number of people immune already by natural transmission, would thus range from 8.3% (if the CDC's estimate of the percentage of 40% of cases going undiagnosed is correct, to 48.8% if the research suggesting 8/9 of cases go undiagnosed is correct -- due to being asymptomatic or having mild symptoms passed off as flu or a cold). In each case my estimate is arrived at by taking the 5.6%, multiplying it by the relevant factor 5/3 or 8/9, and subtracting the fatalities.
If we assume that being naturally immune and being vaccinated are statistically independent events (a conservative assumption since we already have healthcare workers who already had COVID and recovered passing up vaccination), a 56.3% vaccination would produce a 60% immunity rate if the CDC's low estimate is correct (even ignoring the increase in natural immunity as the pandemic continues to rage), while a 22% vaccination rate would give a 60% immunity rate of the 8/9 estimate is correct. For each 10% increase in the immunity rate needed to extinguish the pandemic by herd immunity, if the CDC is correct an additional 10.9% of the population would need to be vaccinated, while if the higher estimate of already attained natural immunity is correct, an additional 19.3% of the population would need to be vaccinated -- these numbers are higher than 10% because the vaccination program will, necessarily be vaccinating the people who had natural immunity due to prior infection (whether diagnosed or not). So, we would reach 70% immunity when 67.2% of the population vaccinated if the CDC is correct, or when 41.3% of the population is vaccinated if the 8/9 estimate is correct.