When you are measuring the specific gravity, you want to know the weight of a specific volume of liquid.
This being said, the refractometer is not strictly speaking a tool to measure the specific gravity. It does measure the refractive index of the liquid. Assuming that you deal with water plus sugar, you can calculate the quantity of sugar from the refractive index, and from this, the specific gravity, so it is really an indirect measurement.
To directly measure the density of a liquid, you can put a precise amount of liquid in a graduated cylinder and weight it to directly get the density of the liquid. However, this is not very convenient as it is often difficult to measure the volume precisely because of the meniscus on the side of the tube. In addition, you need a precise graduated cylinder, a precision scale and maybe a pipette, which is not very convenient. A way to do so precisely is with a pycnometer what you call a "bottle to measure specific gravity". You still need a precision scale though
Using Archimedes principle is in the end the most convenient way to measure gravity, as it can be done with a single, inexpensive tool, the hydrometer. So you mentioned the main tools already.
Regarding units, tools that measure density or specific gravity will either give results in g/cm^3 or as a uniteless ratio of the density of the liquid over some reference liquid (such as water). One could argue that hydrometer give a result in °P, but this is not exactly true. Hydrometer directly measure the density of the liquid, but because they are mostly used in brewing or wine making, you find them with a °P scale, so you don't have to make the conversion (although we often do it backwards to go back to specific gravity from the °P result).
Now they are other methods to measure the density of fluid, but they are often hardly available to brewers. You can use ultrasonic waves to infer the density (as speed of sound in a material depends on this parameter). You can also do it by measuring the pressure inside a column of liquid as a function of depth, etc.