Are these through holes belong to buried via?

As stated in the topic, my team made a flex-rigid PCB then send it to fab. There is no copper in the down side parts of the board, but there are vias in there. We think this are through holes and no buried vias, but the PCB fab told us these are the buried vias and need $100 extra fee. We are confused about it. Are there somebody meet the same problem? How would you fix it?
great thanks

do you have the gerbers? or fabrication notes? or even a picture of that PCB? you have to make sure before you claim anything!

Whether there's other copper artwork on the bottom layer or not, regular through vias need pads on both outer layers. Do you have it? If not, why?

If you don't want the via copper exposed to the bottom, you might consider backdrilling instead of buried vias.

Flexi rigids are quite complex to manufacture and ALL holes require a copper land on ALL layers to make them reliably, otherwise cost and yield rates go through the window... I would suggest getting some more information on designing flexi-rigid PCB's and talk to possible manufacturers before you start designing to get all the requirements and parameters set up first to avoid costly mistakes.
I do a few flexi-rigid's and recently completed a 14 layer (12 flexi layers, all 2oz copper) where the design was quite hard due to the manufacturing requirements of these types of PCB's (and to keep costs down, the bare PCB price makes your eyes water), and in the new year I have to shrink it down into less space...
Start by downloading and reading this book, its free and a good primer...
https://flexiblecircuittechnology.com/flex4/

Plenyt more info out there from the actual PCB manufacturers.