As a powered lift truck technician who specializes in electric lift trucks and vehicles I have some knowledge and experience on this subject.
The four LED display in your TT is quite simply a limited range voltmeter. It likely ranges between 12.8V to 12.0VDC. As you've already guessed, it's not terribly useful.
If you're serious about battery maintenance, a hydrometer coupled with an inexpensive voltmeter is the ultimate solution.
A hydrometer will run about $10 and another $10 for a cheap voltmeter are a much better indicator of battery SOC (coupled with good documentation) than ANY other meter. Perfect for your 6V FLA batteries. They require good documentation be kept so you'll have reference of the past (SG/cell and battery voltage need to be kept, good notes will indicate everything from acid imbalance to cell failure), they're also messy which leads to inconvenient.
You can buy a toaster (coil type battery load tester) for about $80, will also provide a decent indication (though pulls off charge with each test) or an electronic battery load tester like a Blue Point YA2612B, but at $200 gets to be pricey, also pulls down battery with each test.
Last but not least, a Curtis 908 BSOC gauge. These are also fooled by being hooked up to shore power, provide an additional 6 segments and cost almost as much as an electronic battery load tester.