How to see if a coax cable is able to receive Internet?
Simply call your cable provider to come hook you up to internet. If need be, they will replace the cable, otherwise they will simply terminate it.

Your "companioned electrician" is misinformed, as are you. You don't need triax or biax for getting internet from the cable TV folks, even though there are other systems which use such cables, mostly hoary antiquities at this point in time. A coaxial cable has two conductors - they are, not surprisingly, coaxial - a center conductor, and a shield, which is also a conductor. Which makes two conductors, not one.

Regardless, just call your cable internet, TV, etc. provider and ask them to hook you up. The vast majority of CableTV internet comes in on exactly the same cable as the TV does, though if the cable is in poor condition due to age or abuse they may choose to replace it anyway. They have specialized test devices the installer will use to check the signal levels at the cable and determine if it will work. Since those are specific to the cable system and typically have a pricetag near that of a car, it's not a DIY project, it's a part of their service provision.