Coaches are identified by 2 things. One is the shell of the coach ( the shape, material etc) and the other is the type of bogie used.
ICF coaches come in all varieties. There are self-generating rakes, and then there are non-self generating rakes which requires EOG. They come in all kinds of liveries. The normal blue, Rajdhani livery, Garibrath livery, JS livery etc. But in the end all of these various classes are identified as ICF coaches because of (1) The use of mild steel for the coach shell and the distinctive shape of the shell and (2) all of these different coaches have the same ICF all-coil bogies.
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Similarly all LHB coaches are characterized by 2 things. One is the Stainless steel coach shell ( this is also shared by the Hybrid LHB rakes) and more distinctively, the FIAT bogie. These bogies are not of Indian design and were imported as part of the LHB rake technology transfer. So, if a coach has both a stainless steel body shell and a FIAT bogie, it is undoubtedly a Pure LHB rake.
Hybrid LHB coaches have a stainless steel coach shell and use ICF bogies. The difference between a hybrid-LHB and pure-LHB is the FIAT bogie and nothing else. And IR officially doesn't even call these coaches as hybrid-LHB. They are referred to as 'Low cost Hybrid stainless steel coaches'. But since it is well known that it is just a mix of LHB and ICF, we call it as hybrid LHB. For all practical purposes, it can be even called hybrid ICF as the bogies are of ICF design.

No, the LHB coaches are not light weight because of HOG/EOG. They are light weight because of the stainless steel used in the construction. The equipment required for self-generation barely weighs 200-300kg, but an LHB coach is usually at least 7 tons lighter than an ICF coach !!This is because of the modified construction and use of lighter stainless steel material for the coaches. And they are high speed coaches because of the FIAT bogies.
ICF rakes running with HOG/EOG have been running on IR for several decades before LHB was introduced. But the they were more or less the same weight as the self-generating rakes.
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These coaches are exactly identical to the original LHB rakes, with every feature. The only difference is the addition of an alternator on the axles to give them self-generating capabilities. No other difference.