By the strictest definitions was Luke Skywalker ever a padawan? If you relax the restrictions, are there any valid canonical reasons for calling him one?
There's no reason to believe Luke was ever a Padawan. He was an apprentice, but the term padawan implies rank in the structure of the Jedi of the Old Republic.
Padawan had specific duties and specific dress codes and marks of station. Luke never had any of the trappings of that Order, never had any of the marks of station, and fulfilled very few duties of Padawan.
Additionally, Luke had never gone through the Trials to advance to Padawan stage, though Yoda did put him through a trial in the cave on Dagobah. Luke failed that test. He was, eventually, given the title of Jedi Knight, and eventually assumed the title of Jedi Master. He never was an Initiate or Padawan, though. To use military terminology, he graduated from an accelerated Officer Training School (going from NCO to the Officer Corps) instead of going through West Point.
When Luke rebuilt the Jedi Order, he reinvented it in a way that did not mirror the Order as it was at the start of the Clone Wars, but as it had been centuries before. One Jedi again could train multiple apprentices, a firm structure was not imposed, and the New Order lacked many if not most of the traditions of the Old.
In short, no. Luke was never a Padawan (except in that it loosely translates to 'apprentice') in the model of the Old Jedi Order. Rather he was a Force-sensitive who became a Jedi through trials of spirit and combat, in the fires of war. He harkened back to an older tradition, and reinvented the Jedi in that tradition.
Later in his life, he did incorporate some of the more useful traditions of the Order (as it was during the Clone Wars), but there's never been a mention of him recreating Padawan in name.