Agglutination test using antibodies
If you have an antibody that is directed against, for example, a bacterial surface protein, then by mixing the bacterial cells with the antibody at a suitable stoichiometry you could observe clumping of the cells as the antibody molecules essentially cross link the cells together. This would be an example of using an agglutination (clumping) assay with a particulate antigen. You can also probably see why it is important that each cell (particle) carries many copies of the antigen for cross linking to take place effectively.

If, however, the antibody of interest is directed against a soluble protein, then mixtures of antigen and antibody are less likely to form cross-linked complexes, although it can happen and can be used for example in the Ouchterlony technique. If you want to use an agglutination test (i.e clumping of particles) with a soluble antigen then you can achieve this by attaching multiple copies of the antigen to macroscopic particles. This is what is being described in the definition of Latex agglutination in your question.

I imagine that there are ways of attaching a soluble antigen to red blood cells (e.g. by chemical cross-linking to the red blood cell surface) in which case you could observe clumping of the red blood cells when you added your antibody.