This is a very common question that requires a long answer... so please search the past postings as well as old pure-silver archive.
The color of the image is primarily determined by the size, shape and surface structure of the developed silver. There are numerous techniques to modulate these factors in emulsion making as well as developer formulation, but most of the knowledge is empirical. Many information printed on amateur darkroom books are also cut and paste work from old books, sometimes out of context.
Warmtone developers are usually made from hydroquinone only, with significant amount of bromide, and pH sufficiently high to develop in 3-10 minutes. These developers require overexposure of the paper and premature termination of development, so it is not easy to make many prints to exact same image density. However, if you want warmtone without toning, this is absolutely the bst way.
"Warmtone" developers containing Metol, glycin, p-aminophenol, Phenidone, Dimezone, etc. are not quite warm tone developer, but they can be formulated to produce warmer side of neutral black, when combined with right paper. The effect is usually much less dramatic than the true warmtone developer. Development is also short, 1-4 minutes range.
Neither type of the developers has warming effect if used with neutral or cold tone paper. They must be used with warmtone paper to be fully effective.
No matter which developer formula you use, if you want good warmtone, it is best to give slight overexposure to the paper and terminate development when you see the right image density. This requires some practice, but this is sort of a required sill to make most beautiful warmtone without toning. It is also useful to use very slow developer, restrained with plenty of bromide.
Coldtone developers usually contain organic compounds that have high affinity to silver halide. A very common agent is benzotriazole, 5-methylbenzotriazole, 5-nitrobenzimidazole, 5-nitroindazole, 1-phenyl-5-mercaptotetrazole (more commonly added to emulsion), 2-mercatoimidazole 6-sulfonic acid and other proprietary compounds. These agents generally retard development, and can't be added too much. Also, many of these compounds are "taken up" by emulsion at a rate considerably different from the rate of taking up the developing agents and water, so that the color of the image may not remain stable over the life of the developer. There are a lot of proprietary compounds to overcome these problems and provide good blue black image, but I doubt that only technologically advanced manufacturers can use such compounds.
HOWEVER, if you want to make warmtone image with polysulfide or selenium toning, it is best to use a warmtone paper and develop it in neutral tone developer rather than special purpose warmtone developer. Many warmtone developers produce prints that become very cold toned, or lose richness of the blac, when toned in these toners. Making warmtone image without toning, and making warmtone image with toning require different kinds of developer formulation. This point was very clearly described by people like Ira Current when he was active in photography, but more recent authors seem to be ignorant of his work or common practice of 1940-1960s.