To answer that, we first need to know when Houdini legally changed his name from Ehrich Weiss to Harry Houdini.

He petitioned the courts for a name change, which would go into effect during his Atlantic crossing.

July 5, 1913 The Billboard page 13

On July 8^th, 1913, he departed NYC on the Kronprinzessin Cecilie (after giving his mother a farewell kiss) and landed in Hamburg about a week later, where he and Bess took a midnight train to Copenhagen.

Now, we just need evidence of his first performance. According to a number of sources, Houdini opened at the Cirkus Beketow in Copenhagen.

16 July 1913 Ad promoting Houdini (Copenhagen newspaper Ekstrabladet)

At 8 pm (CET), Jul 16^th, 1913 Houdini performed in Danish, before a circus audience at the Circus Beketow in Copenhagen that included the princes of the royal family.

And then at noon (CET), the next day at a Press reception, Houdini received news of his Mother’s Death, who had passed away 15 minutes passed midnight in Asbury Park NJ, which was 6:15 am (CET) in Copenhagen.  The shock of Houdini’s mother death had been followed with an attack of a “chronic kidney disease”  and Houdini was released from his contract and returned to the states.

However, the performance in Copenhagen was not his first appearance with the legal name of Harry Houdini.  You see, he performed at Sea on July 12^th 1913 at 9 pm as evidenced by the following program:

Swann 2006 Catalog Lot 166 – Program for a performance by Houdini onboard the Kronprinzessin Cecilie.

For reference and additional information, please refer to the following sources used in this post:

• Houdini: the Copenhagen Affair by Johan Ahlberg (courtesy of Handcuffs.org)
• Houdini The Career of Ehrich Weiss by Silverman, pages 179-180
• Houdini The Untold Story by Christopher page 132