This excerpt from Wiki explains why there are 2 different celebrations of 125 years:
The first steam locomotive railway line was opened on 15 July 1846 between Pest and Vác. This date is regarded as the birth date of the Hungarian railways. The Romantic poet Sándor Pet'fi, who later became a leader of the Hungarian Revolution of 1848, rode on the first train and wrote an occasional poem, predicting that rails would connect Hungary like blood vessels in the human body.
After the failed revolution, the existing lines were nationalized by the Austrian State and new lines were built. As a result of the Austro-Sardinian War in the late 1850s all these lines were sold to Austrian private companies. During this time the company of Ábrahám Ganz invented a method of "crust-casting" to produce cheap yet sturdy iron railway wheels, which greatly contributed to railway development in Central Europe.
Following the Austro-Hungarian Compromise of 1867 that created the Dual Monarchy of Austria-Hungary, transport issues became the responsibility of the Hungarian Government, which also
inherited duties to support local railway companies. This came at a considerable cost: in the 1874 fiscal year 8% of the annual budget was spent on railway company subsidies. This led the Hungarian Parliament to consider founding a State Railway in 1868.