After spending some time today investigating here is what I think:
Strike: Main line combat. Largely a generic distinction incorporating a healthy amount of medium/heavy/assault mechs with elemental and aero support. Some Strike Clusters seem to be heavier weight. All appeared to be focused on ground combat. Akin to an assault/battle/Guard Cluster in other Clans. Should not be confused with the "Striker" designation in some Clans that indicates a lighter weight unit.
Cruiser: Oriented toward speedy mechs, light and medium weight machines, many have ample aero or elemental support to offset their lack of staying power. These Clusters emphasize tactics and strategies that emphasize speed and maneuverability and some are noted for their scouting skills as well. In some other Clans these units would be called Striker or Recon Clusters.
Combined-Assault and Combined-Strike: The exact composition varies depending on Galaxy Assignment and the availability of an Air Assault Cluster. In Front-line Galaxies where no Air Assault Cluster exists, Combined-Assault or Combined-Strike Clusters emphasize air support with multiple ASF Trinaries or Binaries and are the central focus of air power for the entire Galaxy. Their structure is similar to some Cloud Cobra Clusters, with multiple aero Trinaries backed to a few mech/elemental Trinaries or Binaries. In Galaxies where "Air Assault" Clusters exist, these units are more varied in composition. With some emphasizing strong Elemental/Mech combos while others emphasize strong mech/aero combos. Or they are evenly split, one Cluster had 2 mech Trinaries, 2 Elemental Trinaries, and 2 Aero Trinaries. The exact composition is molded in part by the needs of the parent Galaxy. These units are sometimes molded to shore up wherever the parent Galaxy is weak. For example in the Rho Spina Galaxy, their Combined-Assault Cluster was assigned the Galaxy's only air support. Oddball elements, like armored vehicles, also tend to end up in these Clusters. Front-line units of this type should be thought of as Air Attack Clusters. Second-line units of this type in other Clans might be considered equivalent to "Regulars" or PGCs.
Air Assault: As the name suggests, these units make use of multiple Trinaries or Binaries of fighters, with sometimes as little as a Trinary or Supernova of ground forces backing them up. They tend to be the second-line equivilant of the air focused Combined-Assault or Combined Strike Clusters in front-line Galaxies.
Command Clusters: Often shaped by the Galaxy Commander as they see fit, these units vary in composition, but they tend to have a healthy amount of every unit type, 'mechs, elementals and ASFs, with Supernovas being common. They tend to get some of the best equipment and warriors.
Its important to note that plenty of exceptions exist. For example in Gamma Galaxy the presence of a mech-heavy Assault Cluster has allowed the 21st Strike Cluster to focus itself into a mostly Elemental Cluster. Some units have also been molded by tradition and their composition has veered away from what they probably were originally, but they retain their original name.