According to the document detailing the plan put out by the Austrian ministry for Social Affairs, Health, Care, Consumer protection, it will be enough to have received the first injection:
Sobald die entsprechende Rechtsgrundlage geschaffen ist, werden auch geimpfte Menschen von der Testpflicht ausgenommen sein. Dies gilt ein Jahr lang ab Tag 22 nach der Erstimpfung.
By contrast, in many countries (and, possibly, to return to Germany more easily), “vaccinated” means having received all a full vaccination course (i.e. two doses for most vaccines).