Trace has nothing to do with the wand (must watch)
The trace is not related to the wand, not related to the wand, not related to the wand. Important things must be repeated three times.
The idea that the trace is related to the wand comes from fanfiction settings; some HP fanfics have this setting, not the original work's setting. This book does not adopt this fanfic setting.
If someone believes this book is 'toxic' for not adopting other fanfic settings, then this book simply isn't for you. Perhaps 'A Bite of China,' 'Top Student,' 'Legendary,' or 'Poisoned Egg' next door might suit you.
If you still find none of them suitable, it's best to temporarily stop reading HP fanfiction.
Note: Many fanfics believe that if you use a different wand, or cast a spell not in your own place, you won't be detected.
Using a different wand isn't because the trace is on the wand, but to avoid leaving evidence, and you also need to make sure that wand isn't found by the Ministry of Magic. (This way, even if accused, you have an excuse to argue, as there's no evidence proving you used magic.)
Case Study:
When students from Wagadou (Africa) are accused of violating the International Statute of Secrecy, they have a powerful excuse: 'I just waved my hand, I never meant to dislocate his jaw.' (Most Wizards never master Wandless Magic in their lives, let alone underage Wizards. But this excuse cannot be overused, and don't use magic in your place of residence (Muggle residential areas).)
The best solution is: stay in a place full of Wizards. (Diagon Alley, Hogsmeade, or a Wizard's home. Remember, it's not just any random place.)
If an underage Wizard is in a place surrounded by adult Wizards, the Ministry of Magic cannot distinguish who is casting the spell. In such cases, it relies on the family's self-awareness, and I believe no one would be so bored as to accuse you at the Ministry of Magic of violating the decree for the reasonable restriction of underage sorcery.
You only get into trouble when you are accused. This is a legal loophole.
In 'Prisoner of Azkaban,' why wasn't Harry punished the second time he used magic? Because Fudge did not accuse Harry. (The punishment here might refer to expulsion.)
In Dumbledore's own words: You were very kind not to press charges at the time. 'Order of the Phoenix.'