Does this mean that if you get vaccinated with AstraZeneca for example, then assuming above numbers, there is 5% chance you will get the virus?
No. {1} gives a clear example to understand the definition of what "AstraZeneca is 95% effective" mean:
A 95% vaccine efficacy means that instead of 1000 COVID-19 cases in a population of 100 000 without vaccine (from the placebo arm of the abovementioned trials, approximately 1% would be ill with COVID-19 and 99% would not) we would expect 50 cases (99.95% of the population is disease-free, at least for 3 months).
Regarding your second question:
Also my question is do these percentages say anything about whether you will develop severe symptoms or not? For example, does it mean that when we say Chinese vaccine is 50% effective does it mean it will reduce the severity of symptoms on half (due to 50%)?
No, the "95% vaccine efficacy" says nothing about whether one will develop severe symptoms. One can read the results of the trials to get numbers on how often participants developed severe symptoms (e.g., see https://www.cnbc.com/2021/04/01/pfizer-covid-vaccine-is-91percent-effective-in-updated-data-protective-against-south-african-variant.html: "The shot was also 100% effective in preventing illness among trial participants in South Africa").
Note that this is the definition used in Pfizer and Moderna trials but nothing prevents someone else to change the definition. E.g. in https://www.youtube.com/watch?v=K3odScka55A, they use a definition of vaccine “efficacy rate” that focuses on the ability of the vaccine to prevent death or hospitalization, not necessarily symptoms or illness. In this context, the video argued that of the vaccines listed, all have the same 100% efficacy rate based on their clinical trials (but not necessarily real world settings.)
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