“There seems to be no clear and explicit definition of what is meant by a “heimeshe hashgacha””
I don’t know what’s giving people that idea. Some people don’t fully trust the mainstream, MO hechsherim like the OU. This doesn’t mean they wholesale boycott the OU like we all do with triangle-k, they just want an extra label from a chareidi organization slapped on certain products in order to signal that that particular product can be trusted. Sometimes it might imply extra supervision for that particular supplier, sometimes I imagine it’s just a label to say “it’s okay to hold by the OU’s supervision on this specific product.”
If you go to a grocery store in NYC, it’s very clear what people mean by “heimish hechshers.” To your credit, I think if I had never been to a grocery store in NYC, I would have no idea what people were talking about either.