What do the reported levels of protection offered by various vaccines mean?

#3  - it means that when they did the trials, and people in the trials reported symptoms and got tested and it was covid, that happened 20x more often in the control group than the experimental group. 

There are 2 ways to interpret this :

• people will be 95% protected from Covid
• 95% of people will be protected from covid

The reality is probably somewhere in between with some people being 100% protected and a small number with much lower protection.

As River mentioned, this was the reduction in reported symptoms during the trials last year. The situation this year is likely to be different due to the variants and also the length of time since people got their jab, with immunity levels likely to decline slowly over time, hence the possible requirement to get a further booster shot in the next 6-12 months

It almost means 3. It means the Vaccine Efficacy is 95%

$VE$ is calculated this way:

$VE=1-\frac{v}{c}$

where $v$ are the number of sick people in the vaccine group and $c$ is the number of sick people in the control group

So if 100 got sick in the control group and 5 in the vaccine group then:

$VE=1-\frac{5}{100}=0.95$

So it's a 95% reduction in your probability of getting COVID :)

Note that the number reported is sometimes the mode and sometimes the mean of the distribution, but beta/binomial distributions are skewed so the mean is often lower than the mode. I have written a blogpost where I redo the Pfizer analysis

See COVID-19 vaccine efficacy and effectiveness in The Lancet:

Vaccine efficacy is generally reported as a relative risk reduction—ie, the ratio of attack rates [i.e. any symptomatic infection] with and without a vaccine.

Ranking by reported efficacy gives relative risk reductions of 95% for the Pfizer–BioNTech, 94% for the Moderna–NIH, 90% for the Gamaleya, 67% for the J&J, and 67% for the AstraZeneca–Oxford vaccines. However, RRR should be seen against the background risk of being infected and becoming ill with COVID-19, which varies between populations and over time.

The good news is that the vaccines (particularly mRNA vaccines) are also very effective at preventing severe disease conditional on any symptoms, and reasonably effective at preventing death conditional on severe disease - so in an important sense, the 95% figure is an underestimate of the relevant risk. In particular, the AstraZeneca vaccine has much better than 67% relative risk reduction of severe COVID.

It would be nice to have better information on prevention of transmission and asymptomatic infection too, but my understanding is that they're good enough that the challenge in reaching $r_0<<1$ is almost entirely the fraction of people vaccinated (i.e. politics and public communications).