If I take a custom made action and build a gun to suit my personal preferences from that or buy a gun from a pawn shop and keep only the action and build a gun from there I see little difference.
I see only a minor difference between those and rebarreling a rifle that I have - if I go to the trouble to have the action trued and the stock modified to fit me a bit better - even if I keep the chambering.
What is a custom rifle? it is one that I have modified to fit me and my preferences better than the off the shelf gun does. It usually requires a barrel that is better than the factory stuff and a trigger that is set to my preferences. I use cartridges that fit my preferences and they are not always available - so a custom chambering is employed but not necessarily a wildcat.
Are these custom guns?
03A3 Remington - action trued, barrel trued and refit - still in 3006. Bolt turned, drilled, tapped, one piece mount and a Simmons scope. Timney trigger and safety and a semi-finished stock, bedded with glass, hand-rubbed Tung oil finish and the barrel free floated.
Remington action trued, Douglas barrel in magnum profile, Timney trigger, factory stock glass bedded, refinished in tung oil, cut and fitted with a recoil pad at a 13" trigger pull. Base and scope mounted and chambered in 358 Winchester.
A Thompson Center TCR chambered for 6mm-30 (30-30 necked down - no shoulder change). Trigger polished and tuned. It came with the scope of my choice. Stock cut down and fitted with a recoil pad to optimized the length of pull for me.
and then this one:
An SKS - taken down to the action. The action was trued, and machined for new cartridge. A Douglas magnum barrel fitted and chambered in 357 Maximum. No gas system, new stock in plastic. It was a lot of work and is a great "straight pull" bolt action 357 Maximum carbine and shoots very well.
For me, they are each custom in one fashon or another but only the carbine is a total remake from the ground up. My favorite for shooting is the 03A3 but the 358 comes in as a close second.
Paul