Power Armor is fairly simple, a suit of combat armor that supports it's own power supply and assists in movement with a variety of techniques and technologies such as artificial muscle 'myomers', conventional hydraulics and electric servos, or some other method of offsetting the heavy weight of the conventional suit.
Light Power Armor has as implied, limited armor protection. Most light armors are used by groups not expecting heavy firepower such as police special forces, on site garrison and security details, and the like. Scouts, recon units, and special forces also favor light power armors as these armors typically have improved mobility compared to standard and heavier armors.
Heavy Power Armor has greater than usual armor protection and tends to be a bit larger and more bulky than standard armor suits. These suits are used in supporting roles, as they have greater potential strength, and can carry heavier loads, larger guns, and bigger gear than regular units.
Aerial Power Armors are designed very similarly to light power armors, but use aerospace grade materials for good protection, and have advanced mobility, coming from jet packs, jump packs, A-pods, or other applications of arcanotech.
Marine Power Armors have water jets and thrusters for moving underwater, and other equipment for their specific environment. In the Cosmic era there are not many marine power armors as submersibles, submarines, and aquatic mecha predominate and the smaller power armors are very vulnerable to concussion and depth charge type weapons.
Space Power Armors are much akin to marine power armors, and are designed specifically for the challenges of their hostile environment, which typically includes heat dissipation, lack of atmosphere, and radiation.
Hardsuits typically differ from power armor in that they are either not designed for front line combat, or are lower base technology. Most hardsuits are going to be found in the civilian and industrial sector where they are used for manufacturing, construction, logistics, and other areas where they function as force multipliers. Industrial and commercial hardsuits are going to be designed for their specific task, such as mining suits having mining equipment, and construction suits having rugged builds and much greater strength than conventional suits.
Hardsuit can be used a derogatory term, degrading all military power armor troopers into the same category as scrappers and metal salvagers.
Avatars sit in the gap between dedicated mecha and large power armors. The difference between and avatar and a power armor is that in a power armor, the arms and legs of a soldier actually occupy the arms and legs of the suit, whereas an avatar pilot is sitting in a cockpit in the torso of the avatar. The difference between an avatar and a mech is scale. Avatars use power armor scale weapons and armor, and not mecha grade equipment.
The avatar is a stop gap that appeared during the development of mecha and power armor as the machines grew increasingly larger. As such it has features in common with both, but it has a number of problems also associated with it. As a stop gap, it is too large to be an effective battle suit, as it is easily confined in small spaces such as urban environments and inside arcologies, and is a large and easy target for power armor troopers. Tactics and weapons developed to work against full size mecha are very effective against avatars, which makes them even more unpopular. This innate weakness and their large size making them first round targets makes them unwanted assignments. On the other end of the scale, avatars fare poorly against mecha, as they are still functionally power armor, they are vulnerable to anti-power armor tactics used by mecha, as well as not faring well against mecha scale weaponry.
While there is potential to have many different versions of avatars, they are generally relegated to terrestrial roles, and most are involved in commercial and civilian operations, demolition, construction, resource gathering, etc.
In poorer parts of the world, avatars are used in place of mecha. These machines are typically made of second line parts and equipment, but make up for this deficiency with numbers, and the support of standard or low tech power armors, and limited numbers of other military equipment.
Author's Note: The information provided here is specific to the Cosmic Era setting, which is in large part not determined by what the best or most exemplary tool or system would be, but what would be used as a function of the society and culture using it. A reference back to the last tank has some input on this. Society has decided that armored boxes with guns are bad, so have banned their use, and have resorted to men in armored suits and giant stompy robots. There are going to be many experiments and failed designs that didn't win for reasons other than mechanical ones. Companies and designs fall out of favor, weaker designs can win through better marketing, and a single disaster can kill and entire school of weapon design.