Indian Railways: Divided By Zones, United By Railfans - Karthik

This is a newspaper article from 1905. You can read the name clearly mentioned in the article. I believe that is more 'reliable' evidence than what we have on Wikipedia.

This train was never known as Punjab Limited. Punjab Limited was a separate train that started in 1927 and ran for one year before getting cancelled. Punjab mail went to Peshawar only for a few years between 1920 and 1930. It used to run via Ambala until March 1st, 1934 after which it was diverted through Feorezepore (I have the newspaper cutting...
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for this as well).

There is nothing significant about 1st June 1912. Punjab mail was started well before that date. IR doesn't care much for heritage or historical accuracy. They celebrate the birthday on June 1st because, a few years ago, some railfans thought that was date of inauguration based on the limited research done till then. Since then a lot more research has been done and now we know more about this train.