We can see that almost all of the servers are interconnected between each others using submarine cables which ends in a complete hierarchy but question is are those internet connected cables are used only for connecting internet servers between each others ?
It depends what you define as the "Internet", If by the "Internet" you mean the large mesh of publically routable IP space then no. These cables can also be used to provide direct private optical waves between two endpoints, similar to a very long point-to-point Dark Fibre.
When these wavelengths are sold to providers it is then up to the provider to determine how they wish to use this service. If they want to run their International IP backhaul over it then so be it. From the Cable manufacturers point of view, they're purely selling a Layer 1 service.
No this cable can carry different types of traffic other than internet traffic . Submarines cables are just physical connectivity from and to connecting two geographical locations . Further to i this cable bandwidth can be utilised for internet and point to point links for private leased lines connectivity this can be achieved by creating vlans isolating traffic that passes through this Submarine cable using Vlans