Hey jvan
It's actutally a good question and likely does pop up from time to time, here's my take on a "custom" knife. For the sake of argument and brevity let's assume minimal CNC utilisation (this opens a whole other can of worms), pure hand work.
The definition of custom has blurred over time, at it's core it has a specific meaning, but often a word is defined by it's common use.
By it's strictest definition a true custom knife is a one off piece designed between the maker and the customer (the customer input adds the "custom" label). I say between the maker and the customer because very often a buyer will find a maker with a style he likes and set him/her to the task of making something special, often allowing the maker a bit of artistic licence and incorporating the maker's own unique style into the knife. There doesn't have to be maker input if the client is specific about exactly how the knife must end up. The important part is the client's input. It may be as little as just the shape of the knife but the customer may go as far as dictating size, materials and finishes, all depends on the project. I feel it's also the customer's decision whether replicas can be made at a further date, the majority of my clients have been more than happy for me to make the design again, but a few have asked that it be kept unique, I have no problem with honouring this at all.
Custom knives tend to be intensive. When I make a custom it all starts with a blank sheet of paper, a pen and a sit down meeting (or a myriad of emails if it's international

). A maker will often send work in progress pics or have follow up meetings just to make sure the project is proceeding the way the client envisages, it's not easy to create exactly what's in someone else's mind, this is why picking a maker you like is important, their own style will fill the little gaps and flesh out the knife.
I feel once the knife goes beyond that first knife for the first owner (ie: is re-made for others), it no longer carries the custom label, it becomes a handmade knife. That said, if 5 friends club together and between them commission a maker to make 5 knives of the same design for them, it could be argued that they are all custom knives. If however the maker is allowed to make replicas of the knife for future sales to new clients, they will not be customs.
The term gets a bit murky if and when the custom knife changes hands. If the initial buyer sells his knife, the person who buys it played no role in the conception of the design and therefore is buying a handmade knife.
As far as sole authorship, very seldom does a maker cover do absolutely everything. If a maker sends a knife out to a specialist engraver to embellish it, it doesn't affect the custom label. I don't however feel that knives made from kit parts deserve the custom honorific.
In terms of runs of knives, if they are handmade I feel from a maker's standpoint that the term "factory" is misleading. It makes me think a lot of it was automated and while there is nothing wrong with automation, it will affect the perceived value. What would you be willing to spend more of your hard earned cash on? And what will you treasure more? Something designed by a master craftsman and made by a machine or something he made with his own two hands and skills? I do batch runs of knives when I can, it helps me get in a groove and you get more work done with fewer mistakes made. So unless you helped design the knife that is being produced in whatever quantity, it's not actually custom.
Gazza