The linked article discusses three types of noise:
Thermal noise: Thermal noise is the noise that we most often refer to when we talk about resistor noise. Another name is Johnson noise. Thermal noise results from the random motion of electrons through the resistor, and is given by sqrt(4kBTR), as stated in the article. Most critically, the rms thermal noise is proportional to the square root of the temperature (Kelvin scale) and to the square root of the bandwidth of whatever is measuring the noise.
Contact noise: It's not exactly clear what the article is discussing here, but he distinguishes this noise by its 1/f characteristic. Noise with 1/f dependence could arise from the current passing through the barriers between the carbon grains in a carbon resistor (which would be why the article recommends other types) or by current passing through the boundary between the metal leads of the resistor and the carbon resistive material. This is the only type of noise that the original article claims will be reduced by using a higher-wattage resistor. This could be explained by the higher-wattage resistor having more parallel paths through different grain or interface boundaries, resulting in noise contributions averaging out.
Shot noise: I wouldn't normally characterize this noise source as being generated by a resistor. Shot noise is fundamental in any circuit, resistive or not, when current is detected. It results from the fact that current doesn't arrive in an absolutely continuous stream, but one electron at a time. Shot noise is likely to be a problem only in extremely sensitive circuits where other noise sources have been very carefully eliminated, or when high current gains are used.
Either thermal noise or shot noise can be characterized as either a voltage noise or a current noise, based on the Thevenin and Norton equivalent circuits:
In any case the resistor noise is injected into the circuit nodes connected to the resistor. Because it is generally a very low level signal, it's unlikely to be emitted as EMI and cause problems in unconnected parts of your circuit, unless, of course, your circuit is amplifying it.
To answer a question from the comments, "white" noise is noise that has constant power density over frequency. For example, if a white noise source produces 2 nV/sqrt(Hz), and we measure it with 1 Hz bandwidth around 100 kHz or with 1 Hz bandwidth around 100 GHz, we'll measure 2 nV rms noise in either case. Thermal noise is a white noise source, while "contact noise", as mentioned above, is not white because it has has 1/f frequency dependence.