На сайте CDC реакции называют local и systemic:

Vaccines are intended to produce active immunity to specific antigens. An adverse reaction is an undesirable side effect that occurs after a vaccination. Vaccine adverse reactions are classified as 1) local, 2) systemic, or 3) allergic (additional information). Local reactions (e.g., redness) are usually the least severe and most frequent. Systemic reactions (e.g., fever) occur less frequently than local reactions, and severe allergic reactions (e.g., anaphylaxis) are the least frequent reactions. Severe adverse reactions are rare (3).

https://www.cdc.gov/vaccines/hcp/acip-recs/general-recs/adve...

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Note added at 25 mins (2023-03-15 09:32:41 GMT)
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Я не помню, чтобы встречалось focal, хотя этот термин тоже будет понятен, все же обычно встречаю local. Для общей пишут и generalized и general, на форуме есть варианты, не знаю, какой вариант более корректный.

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Note added at 31 mins (2023-03-15 09:38:49 GMT)
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https://www.proz.com/kudoz/russian-to-english/medical/314267...

Здесь, например, local и general, хотя это для пациентов статья:

https://www.seattlechildrens.org/conditions/a-z/immunization...

В WHO, опять же, local и systemic:

These information sheets include a short summary of the vaccine. They also comprise details of mild and severe adverse reactions (local and systemic) following immunization. Expected rates of vaccine reactions have been included if available in published literature.

https://www.who.int/teams/regulation-prequalification/regula...