The title of the press release is "New CDC Study: Vaccination Offers Higher Protection than Previous COVID-19 Infection". That is false. The first paragraph of the release says "These data further indicate that COVID-19 vaccines offer better protection than natural immunity alone". That is false.

Seems like you stopped reading . The full sentence you quote is “These data further indicate that COVID-19 vaccines offer better protection than natural immunity alone and that vaccines, even after prior infection, help prevent reinfections.”

Reinfections is the key word here indicating the study is about reinfections. You’re assuming it’s about initial infections. If they didn’t repeatedly say the word “reinfections” in the press release, you might have a point, but the reality is the CDC summary is accurate, as is the headline message that people will get stronger protection if they get vaccinated.

If someone says "X and Y", and X is false, then what they say is false.

You're arguing that someone who already knows what the study says can re-interpret the statements in the press release as being true, by adding a few additional qualifiers like "among the already infected" here and there, on the assumption that they were just omitted for brevity. But the intended reader, who doesn't already know what the study really says, will read the statements as written, and will receive a false communication.

No, I’m arguing your complaint is based on your own misreading. Some form of “reinfected” or “reinfections” appears 4 times in the press release.

There’s no need to “add qualifiers” - simply to not remove them.

Read the title. What does it say? What it says is false. Note that many people won't read beyond the title.

Read the second sentence of the first paragraph. It says two things, the first of which is false.

You're arguing that the press release is fine, because although it contains false statements, it also contains some true statements.