I think it does mean that the virus spreads slower (although the exact amount will also depend on factors like how vaccinated people are distributed among the population). Whereas in an unvaccinated population, each infected person might spread the virus to (say) five others, in a 70% vaccinated population, on average 3.5 of those others would be vaccinated, so the virus would only spread to 1.5 others. This gives a slower rate of growth. Once that rate of growth falls below 1, herd immunity is achieved. When you are close to herd immunity, that rate should be just a little above 1, leading to slow but positive growth.

I think that the question isn't slow or fast, it's exponential or not exponential. If you are getting 9 infections for every infection then that's a bit exponent. But, 2 for every infection is also an exponent - just a smaller one. Once you get below 2 for every infection then you are in the territory of slow growth - but not for long and not everywhere. The UK had a patchy delta wave, my guess is that when the non herd immune demographics and areas started to get herd immunity (drop below 2) then the breaks really slammed on.