Sure it does, go over the rating and it will burn open which is what it is designed to do. It also gives you an idea of the maximum current that is "Safe" for the circuit.
Auto batteries are charged at a relatively low current compared to what they can supply. Typically they are used to start you vehicle and are rated in Cold cranking amps. These are typically in the hundreds of amps, thus the big wire. But this is for a short period of time. The alternator in my truck can supply over 250 amps and it is really the work horse when the truck is running supplying the current to operate everything. It recharges the batteries and supplies the operating current to all the electronics along with the battery but mainly from the alternator when running.
When you turn your engine off, the battery is the sole source to operate anything electronic and that is where the amp hour rating comes in. I'm probably preaching to the choir but this whole thread started about an inverter that would not operate over a couple of hours, even when the truck is running. I contend there is another issue beside needing more batteries. See my post above about how long mine ran today with and without the truck running.
You can't just take a meter and measure the current at the connector. You need a load to draw current and in the case of the battery it will change based on how much the battery is charged. I'm sure you have used a battery charger and they typically have a meter that shows charging current.. When you first connect it it may jump to 10 amps or more but it drops rapidly and will continue to drop as the battery charges.
You probably will measure a low current at the connector if the battery is connected but that does not mean that that is all the current available. Like I said earlier a 25 foot 18 awg wire can handle 10 amps +.
Make sure you have an amp meter that can measure amps. You will need to put it in series with the circuit which will be a little difficult. Don't let your meter become a fuse.
let me know what you find out..