Is there a gray area between scrumping and foraging?

The distinction between scrumping and foraging has to do with where the collecting takes place. Scrumping takes place on a private property or a property which clearly belongs to someone e.g. a farm, an orchard, a private garden, an intentionally planted out street verge. In contrast foraging is collecting food from the wild, the bush or abandoned places when you don't know the owners.

Scrumping is closely related to gleaning. It's collecting something that the owner doesn't mind you "recycling" or utilizing. Often the owner will turn a blind eye to you collecting it - either because he has surplus, or he doesn't want it or because he knows you need it. Ruth in the Bible gleaned wheat from the fields out of her need - she was trying to support herself and her mother-in-law Naomi (Ruth Chapter 2). This type of gleaning/scrumping is still a custom in Syria - farmers intentionally leave part of their crop unharvested for the poor and foreigners to collect.

Foraging generally occurs in public open spaces, national parks, crown land, on street verges, near railway lines, on vacant property or in abandoned settlements.

In the case of the example you gave I'd call taking the berries from the neighbour scrumping because she knew the owner. Taking berries from multiple gardens is still scrumping because the gardens are clearly maintained private properties. If the berries had been gathered from an abandoned settlement where no one lived, that would be foraging. If someone feels generous and puts a sign out that says "free berries - pick them yourself!" collecting these berries is scrumping. If their motivation for putting out the sign that says "free berries - pick them yourself" is to help out the local poor people, then that's gleaning. Taking overhanging berries that are growing outside the perimeter of the neighbour's property - including berries that are spilling over on your side of a shared fence could be scrumping or gleaning. Taking berries from a bush inside the perimeter of your neighbour's private property is stealing unless permission was granted previously.