When I search for Paris to Vernon on SNCF's website (sncf.com), it shows to take Intercites 13103. When I do that same search on Transilien's site (transilien.com), it show to take TER J (dir Vernon). They both leave at 8:20 and arrive within 1 minute of each other. Are they the same train and just coded differently on the 2 sites? If not, do you have a suggestion as to which one to take? Would you board them a different places in Gare Saint Lazare?
Intercites is long-distance, Transilien TER is local. It's probably a coincidence that they leave at much the same time. They will be from different platforms, but Saint Lazare is a big station anyway and I don't think either will be more convenient. The intercites train may be slightly more comfortable.
The only train the Bahn found at that time was 13103. It is for Rouen-Rive-Droite and only calls Mantes la Jolie at 8:53, arriving Vernon(Eure) at 09:05.