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US Citizens traveling from MAD - VIE, VIE - MAD

Hi lovely Travelers!
We are US citizens visiting family in Spain and in Austria next month. We will all travel to Spain, then during our time in Spain we will fly round trip from Madrid to Vienna.
We are all vaccinated, have recent booster shots, and have our CDC vaccination cards.
We are very clear on the requirements to enter Spain ( I was there last year and we will have the QR Codes for entries to Madrid during this trip)
From what I read on the Austrian consulate page, my understanding is that we only need to show our CDC vaccination cards in order to enter Austria. That seems too my understanding correct?