oh huh didnt know that the taira clan is also known as heike.. where should i go read up the difference between clan name usage vs family name usage???
Read more in detail... I'm not sure, I picked it up over time with the language, so I'll elaborate:
In short, you have a word for “[name]” and others for "house of [name]". Similar to 'Oktoberfest' being a single word that you don't change to "oktoberfest festival", though of course 'october' & 'fest', are much more obvious for English. Which is why you'll see ‘Taira clan’ a lot, because English speakers don't see 'Heishi' and go "sounds like Taira clan to me".
"Minamoto" and "Taira" are 'nobility surnames', used by people from those clans. A lot of historical figures are named "Minamoto no...", "Taira no..." referring to the clan/lineage itself.
Nowadays Japanese surnames function akin to 'given names' in 'western culture', but before the Meiji Restoration (1870) surnames were used similarly to titles and mostly by nobility and the like, to indicate their status/rank. A bit like European nobility, but often just a single surname.
So while these days you address people formally by their surname, when it comes to these historical figures, either just the 'name' or the full name is/was used, since you have hundreds of 'Minamotos'. "Yoshimoto" for Minamoto no Yoshitomo, "Yoritomo" for his son Minamoto no Yoritomo, etc.
The clans were made up by smaller clans / families, but all four major clans of that era (Minamoto, Taira, Fujiwara and Tachibana) included 氏 ("clan" or "family name/lineage") as their name. And the different branches within would use that name in addition with a name of a notable ancestor, like the Seiwa Genji (清和源氏) - the most powerful line of the Minamoto clan - descended from Emperor Seiwa.
In comparison 家 ("family" or "house") wasn't really that common to use, but it has certain undertones for 'distinguished nobility'. 公家 (kuge) refers to the aristocratic class of the Imperial Court in Japan, during the period in which these clans ruled.
It's mostly "the Tairas/Taira clan" (平氏) that gets referred to as the Heike (平家), especially due to Heike Monogatari, where Taira no Kiyomori is one of the central characters of the story. After becoming head of the clan, he gained a lot of power and influence, eventually obtaining the position of chief of government. Then he attempted a coup which resulted in the war. Him and his inner circle (family & clan branch) are the ones to which the Heike name especially applies to, but given their significance within the clan, 'the Heike' often alludes to the whole clan itself.
But again, the term existed before them for a variety of things, from people from the clan that worked as government officials, to vassals of the clan during their rule. Likewise, Genke (源家) is hardly used but does not refer just to the Seiwa Genji.