The adjective berrichon appended to the name marks this kind of meat pie as having originated in the Berry province of central France. (Historically, Saint-Aignan was known as Saint-Aignan-en-Berry, but not any more...)
What distinguishes pâté de Pâques from other pâtés en croûte, and makes it an Easter tradition, is the hard-boiled eggs that are baked in the pie along with the meat filling. The meat is a mixture of ground pork and ground veal. Some versions also include some ground duck meat. Here's a recipe.
It's interesting — to me, anyway — that the dictionary lists the primary meaning of the word pâté as originally meaning pâte en croûte. I think for most people nowadays the primary meaning would be what we also call pâté in English — a meat "paste" baked in a dish that the French call a terrine.
In other words, pâté de campagne, for example — the baked-in-a-dish meat mixture that's served cold, often with little gherkin pickles (cornichons). That's the second meaning of the term pâté in the dictionary I use (CNRTL).
Now I understand why the Cornish/Welsh meat pie is called a "pasty" (pronounced pass-tee, not pay-stee). Even though in Great Britain the meat used is beef, and the recipes includes vegetables in the mixture, it's a pâté en croûte by another name.
What I take away from this is that the pâté de Pâques is an old-fashioned concoction. The Berry province is rural and agricultural, meaning old-fashioned and conservative. I don't remember ever hearing about or seeing pâté en croûte with œufs durs baked into the pie in Paris, though I'm pretty sure you could find pâté de Pâques there these days.