Since I2C, SPI, etc all communicate bits in some serial fashion, is there an equivalent name to refer to the type of communication/protocol that occurs on the TX/RX pins (associated with UART) of a micro-controller other than "Serial" regardless of whether its voltages are TTL (3.3V,5.0V) level or RS-232?
The best answer is "TTL-Level RS-232-Style serial communications" Sometimes with additional specifiers (parity, no parity | 7/8/whatever bits per byte | 1/2/1.5 stop bits, etc...), but those options are defined in the standard..
RS-232, EIA-232 and TIA-232 are all the same protocol, in general. There may be minor differences in the specs, but they generally interoperate without issue. Mostly, it's just an issue of which standard body you're listening to.
In general parlance, "Serial", without additional specifiers is generally assumed to mean asynchronous serial with data patterning following the RS-232 standards, though the signaling levels can vary (generally either TTL or -12V - +12V).
Saying something goes through the UART tells you nothing, unless a specific UART on a specific chip is specified. There are UARTs out there that do all sorts of interesting, non-traditional-serial protocols.
Saying "UART Communications" is similar to saying "Vehicle Travel". Most people would probably think of a Car, until one guy shows up in a plane, the next guy a boat, and one random wacko in a giant hamster ball. They're all still vehicles, and as such all technically correct.
You should never make assumptions like that unless you know the person who says "UART" well enough to feel confident [s]he means what you think [s]he means.
You almost have the answer in your question. It's full title would be "Duplex serial UART Communications"
UART means "Universal Asyncronous Receiver Transmitter".
Duplex because it supports either alternating or simultanous data flow directions.
Sometimes an 'S' is added - USART - when synchronous protocols are also supported. Some UARTs support more than one protocol over the same pair of pins depending on how they are configured. For example HDLC.
There are many synchchronous and asynchronous serial protocols, one of the most common is 8 Bits, No parity, one stop bit. (8N1)
As you correctly state in your question UART defines the data transport protocol, not the physical layer, the voltages or wires. Your question specifically asks about the physical link or pins comparing them to similar pins that support other serial protocols.
There is some confusion about the use of the indusctry standards RS232 or ANSI/EIA/TIA-232 (or for that matter many other physical interfaces or higher level protocol specifications) All of which talk about the signals after the pins or the way the data is interpreted at a high level.
This citation states the UART was coined by IBM in the 1960s.