Good question! It actually becomes clear when you look at where the terms came from.
For a long time, there were only manual transmissions. When automatic transmissions first came out, they were considered a luxury. Hence the manual transmission that everyone knew and loved was called a standard transmission, as opposed to this fancy new contraption. The term standard also refers to the fact that even today, when automatics are vastly more popular in the U.S., a manual transmission is still typically standard equipment, while an automatic transmission is an option you pay extra for.
So standard transmission does mean manual transmission, although the term standard is more commonly used by older people. It can be confusing as there are now cars where an automatic transmission comes standard, so I prefer the term manual. It is also known as a classic transmission, old school transmission, stick shift, dick stick, or simply "stick."
Automatic transmissions, even when they're the only transmission available for a particular car, are never called standard transmissions. They may be called modern transmissions or regular transmissions. Some let the driver select the gear with a stick or paddle shifters, but they are still considered automatic because there's a torque converter instead of a clutch.
5-speed refers to the number of gears, not counting reverse. A 5-speed car could have either an automatic or a manual transmission.
If someone says "I have a standard transmission," they are saying they have a manual transmission (could be any number of gears).
If someone says "I have a 5-speed," they are stating the number of gears (could be either type of transmission).
If someone says "I have a 5-speed manual," they are specifying both the number of gears and the type of transmission.