Is the baby really an "archduke"? [...] I mean, Austria doesn´t recognize the Habsburgs as "imperial" or noble any longer. Other than Germany Princely titles, be it just part of a name, are not recognized (not even a "von") in Austria. The last Habsburg Crown Prince Otto (member of the EU Parliament) was called "Otto von Habsburg" in Germany, while it was just "Otto Habsburg" (or, when being addressed personally "Herr/ Mr. Habsburg") in his homeland.
I agree with you, but nonetheless, the Belgian royal family and government consider Prince Lorenz and his descendants to have, in addition to their Belgian titles, the title of Archduke/Archduchess and the surname "of Austria-Este (Habsbourg-Lorraine)". For example, Lorenz was named as "Prins Lorenz, Otto, Carl, Amedeus, Prins van België, Aartshertog van Oostenrijk-Este" in
the Staatsblad in 2002.
Isn´t he rather a Prince of Belgium?
Technically speaking, the baby is not a Prince "of Belgium" but he is a Prince of the Belgian royal house, as King Philippe chose to reserve "of Belgium" to children and grandchildren of sovereigns and future sovereigns in 2015 while allowing other descendants of Leopold I to remain Princes and Princesses of the royal house. Refer to article 4 of the Royal Decree of November 12, 2015 and article 5 of the Royal Decree of July 12, 2019.
12 November 2015. - Koninklijk besluit betreffende de verlening van de titel van Prins of Prinses van België
Translation:
Article 1. In the public and private acts relating to them, the Princes and the Princesses, children and grandchildren, born in direct descendance from the King as well as the Princes and the Princesses, children and grandchildren, born in direct descendance from the Crown Prince or the Crown Princess carry the title of Prince or of Princess of Belgium following their first name and, so far as they carry them, their family name and their dynastic title and ahead of the other titles to which their ancestry gives them the right. Their first name is preceded by the predicate His or Her Royal Highness.
Article 2. In the public and private acts relating to them, the Princes and the Princesses, children and grandchildren, born in direct descendance from His Majesty King Albert II carry the title of Prince or of Princess of Belgium following their first name, and, so far as they carry them, their family name and their dynastic title and ahead of the other titles to which their ancestry gives them the right. Their first name is preceded by the predicate His or Her Royal Highness.
Article 3. The Princes and the Princesses who already carry the title of Prince or of Princess of Belgium pursuant to the Royal Decree of 14 March 1891 qualifying the Princes and Princesses born in direct male descendance from His late Majesty Leopold I to be Princes and Princesses of Belgium retain this title following their first name and, so far as they carry it, their family name and before the other titles to which their ancestry gives them the right. Their first name is preceded by the predicate His or Her Royal Highness.
Article 4. The Princes and Princesses, born in direct descendance from His Majesty Leopold, George, Christian, Frederick of Saxe-Coburg, who are not covered by Articles 1 to 3, carry following their first name and, so far as they carry it, their family name, the titles to which their ancestry gives them the right.
12 Juli 2019. - Koninklijk besluit houdende vaststelling van het wapen van het Koninklijk Huis en van zijn leden
Translation:
[...]
Art. 3.
The Duke or Duchess of Brabant uses [a description of his or her coat of arms].
Art. 4.
The other Princes or Princesses of Belgium of the male-line and female-line descendance in direct line from His Majesty Leopold I use [a description of their coat of arms].
Art. 5.
The Princes or Princesses of Our Royal House who are not covered by article 4 use [a description of their coat of arms].
[...]