Your house will be grounded either by the metal water pipes as you said, by a ground rod, or rebar in your foundation (if it has it).
The most common modern method is the ground rod. Normally if a ground rod is used, it would be placed on the outside of your house on the other side of the wall your electrical panel is. Because you state that it goes below the floor, it's not likely you have a ground rod (at least not one you can see outside). You can easily check for an exterior ground rod by looking outside next to the foundation on the other side of the wall of the electrical panel. You should see a copper wire going into the ground, and possibly the ground rod itself sticking up a little bit.
You stated that your house is older, so I'm not confident that your house is grounded by any rebar in the foundation as your house probably doesn't have it. If it's grounded to the rebar, you may not be able to observe the ground connection without digging if the foundation was poured over it.
If the panel is grounded to the water pipes, and the ground wire can be seen going into the floor, I'm assuming the side of your house the water comes into is on the same side of your house that the electrical panel is on. I say this because you want to ground the water pipe as close to the entrance of your house as possible. This is so that if, for example, you replace your water lines in the house in the future. If you take off the water pipe that the ground wire is connected to, you will have an ungrounded house. By having it near the entrance, you are that much less likely to accidentally disconnect your ground. I would have thought that they would want the water line to come through the wall of your foundation, rather than the floor, but it's possible.
So how do you determine if your house is grounded with the water pipes? You can do it by observation as described above, but this won't help if you can't see where the ground goes as you can with a ground rod. Some might say to disconnect the ground conductor from the panel and measure the impedance between the ground wire and your pipes. This won't work because the ground path could be going through rebar, a ground rod, or some other method before reaching the pipes.
Unfortunately unless you dig through your floor, you can't tell where it connects. You would need specialized meters that would test ground impedance, but even this method isn't guaranteed to pinpoint the ground connection. An electrician should have this meter though. If I was in your position, I would call my utility company (water and electric) and ask how you should determine the ground connection. Explain that your ground connection isn't visible and that it would be an expense to determine the ground location. Hopefully they will reimburse you in some way, but normally the utility's responsibility ends when it enters your house.