Half way right. Unfortunately, the terms have some historical baggage. No one would name today's gasoline formulation "lead free" unless older formulations had contained lead.
First there were dual supply opamps that were not called that, because all opamps were about the same, with input common mode range and output swing not getting very close to either supply rail. Usually these were operated with dual supplies, but would work with single supplies if the signals were not close to ground or the positive rail. I remember when analog systems almost always had +_12 or +-15 volt rails.
Then TTL logic came along, and the +5 volt supplied system became the most common one. Some people wanted to add a few analog features to TTL systems, and it added a lot of cost to add +- 15 volts to a 5 volt system, just to add a couple comparators or opamps, so the low voltage capable, ground rail inclusive designs were invented (the LM324 series of opamps and the LM339 series of comparators, especially). These let you add analog features to a 5 volt system i.e. the inputs and output stayed between +3.5 volts and ground. The advertising code words to distinguish these new variants of opamps and comparators was "single supply", since they could be shoe-horned into 5 volt systems.
There were no rail-to rail designs for years, until CMOS processes matured for quite a few years and were refined to be used for analog chips. The wasted 1.5 volts of the supply range was not a serious problem for many designs, till the supply started dropping from 5 volts to 3.3 and below. Even though "single supply" designs were improved to function better down to 3.3 volts, the remaining 1.8 volt or so swing was getting to be a real problem, that was pushed back by coming up with parts that really worked with input and output voltages over the full supply voltage, based on low voltage CMOS processes. The advertising code words that have been used to distinguish these new features from the older two types was "rail-to-rail" inputs and outputs. There are also some versions that have only "rail-to-rail" inputs or outputs but not both. This is a gotcha you have to watch out for, when you pick an opamp that is advertised as just plain "rail-to rail".
As far as I can remember, the original dual supply opamps only started being labeled as such only after "rail-to-rail' designs became available, and there was a need to distinguish them from the newer designs by their power supply head-room requirements.