I agree that most people do not understand the math. Here is my effort to explain in a way that most people may understand.
First there is a difference between between efficacy and effective as the article mentioned but they are similar.
Efficacy is measured in a controlled often clinical setting where subjects are, randomly divided into two equal groups. One group gets the vaccine and the other group gets a placebo injection (no vaccine). Then a certain time is waited for exposure to covid.
If both groups were 100 people and 2 in the vaccination group and 10 in the unvaccinated group were diagnosed with covid then an efficacy number is calculated. 2 vs 10 means that there were 8 less covid among the vaccinated and the efficacy is calculated and expressed as 80%. This means that you have an 80% lower chance of getting covid if you are vaccinated compared to the unvaccinated.
Effectiveness is measured in the same way to show how well the vaccine works in the real world. It will often be different from the controlled experiment efficacy number.
A problem with using these terms and numbers is that many people relate this to the individuals percentage chance of infection. They mistakenly believe that with a vaccine that is 80% effective they will only have 20% chance of getting infected. What it means is that the vaccine has reduced your chance of getting ill with a particular covid variant compared to the unvaccinated by 80%,
I do not think that there is some conspiracy theory behind reporting these numbers. More education on the meaning of the number might help. The bottom line is that they fundamentally show the effectiveness of the vaccines and the reduced chance of getting covid for vaccinated vs unvaccinated.