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.

Punjab Mail
The Punjab Mail runs between Bombay and Firozpur. This was the GIPR train; there was another train of the same name that ran for a while between Calcutta and Delhi on the East Indian Railway. The Punjab Mail made its debut on 1st June 1912. Like the later Frontier Mail, the Punjab Mail too used to connect with the P&O steamships on fixed mail days and would steam off from the Mole Station; on other days it departed from Bombay's Victoria Terminus.

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a brief period, an extended service called The Punjab Limited operated between Bombay VT and Peshawar, on the GIPR and NWR; this was a rival to the Frontier Mail, but does not seem to have lasted as a service for long. (There is some doubt whether the Punjab Limited was an entirely separate special service or a special extension of the Punjab Mail.) The Punjab Mail was among the fastest trains in pre-Independence India (probably the fastest one at various times). The train had air-cooled cars in 1945.

It was hauled by a variety of locos. XC locos were used after the rake was extended by the addition of third-class cars in the 1930s. In 1929-1930 EA/1 electric locos were used experimentally. The train later ran electric-hauled until Manmad, where a WP took over. From 1968 the train was diesel-hauled until Jhansi and by 1976 or so it became diesel-hauled all the way. A WCAM-1 loco was used a few times in an attempt to provide continuous haulage without locomotive changes, in the 1970s. Since then, and continuing today, it is hauled by a DC locomotive until Igatpuri and an AC locomotive thereafter towards Delhi and Firozpur.
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