You are correct that those are edge choices, not stone thicknesses. If you purchased a 6cm built up edge, he needs to remove the island top and redo it. Your contractor will need to build up the island subtop by that 3cm and take into account the consequences of that extra counter thickness, like the additional counter height that it adds to the design.

Overhang supports are created in the design phase and installed by your cabinet installer. That is not the stone installer responsibility. Support must happen before the template happens. 3cm needs support beyond 10”, so yes, you'll now have to retrofit something external like L brackets with braces, and likely reinforce the cabinet interior to accept it. Or, if the island is getting redone, that’s the time to have the designer reengineer the support correctly. Which they should have already done.

"On the phone, the owner initially agreed that pencil edge was just an edge profile and not thickness, but he changed his tune the next day, saying that we changed to 3cm by confirming our pencil edge selection."

He's a liar and you can tell him I said so.

A pencil is the worst possible edge selection; it is a chip factory, particularly at sinks. Ogee is workable as long as they don't drop the bit far enough to create a sharp 90* edge at the top.

3cm engineered stone may cantilever up to 15" without additional support, however, over a longer span, more than 6' or so, it may sag. It's not going to break even with all your kids hanging off it.

In regards to the cantilevered edge (12.5” for island and 13.5” for peninsula) is something like this adequate for support if we want a more streamlined look?

As far as our carpenter, he met the fabricator and asked is there is any additional support they want added in and even offered to add brackets himself as he was worried about damage to the cabinets. The fabricator/owner declined and said they would add metal support and they do it all the time and had it under control. Of course, the fabricator denies this, but at this point we just want to add correct support if needed and move on.

Here is a pic of our unfinished cabinets with countertops in place

Chantelle:

"3cm needs support beyond 10” When ABC said this he was correct if speaking about 3cm natural stone. When you said you had "quartz", I assumed you meant engineered stone, not natural quartizite. 3cm estone can be cantilevered up to 15" without additional support. This is the opinion of the estone manufacturers. I might change my mind if I saw your dimensioned layout, but apparently no additional support is necessary.

It is quartz and not quartzite