Myouji 苗字 the Family Name

The family name is what we most often think of as a "byname" or a "surname" in Western naming concepts. 

Kamei 家名 the House Name

This is also a surname/byname type, that indicates what (large, communal) house you live in, whether that is with your nuclear family or not. Thus, it is literally a house name, used as a surname, particularly for lower class families who did not have/were not allowed to have a myouji.

Uji 氏 Clans

In a historical Japanese context, clans are large family groups, but are not necessarily blood-related families.  Clans did not change names after Nara era established some new ones, so picking from a list of existing Uji.  A list can be found under the "Ancient & Mediaeval Clans" table in NCMJ, or this Wiki list of Japanese Clans. 

If you are adding the clan to the name, it is mentioned after the family name, as a descriptor, such as "Takeda family of the Fujiwara clan."   This is why, when omitting a family name, you must indicate the "of the" when saying "Gentarou of the Minamoto" to ensure clarity. This is done with the particle "no." 

Yobina 呼び名 the Spoken Name

"General use name other than the nanori." - NCMJ

Literally translated as "common name," this name is what functions as a given name, that would be used most often/almost always as the name of a person. 

Tsuushou 通称 the "Vulgur" Name

"Name used in everyday conversation" - NCMJ. As far as I can tell, and from the meanings being quite literally the same - the Tsuushou and the Yobina are almost interchangable.  They also follow such similar naming construction patterns that I don't think there's a difference.  I will update this if I learn more. 

Kamei/Kemyou 仮名 the Informal Name

Nanori 名乗り the Official Name

"Names used in proclamations and self introductions."  Commoners do not have Nanori.  By special permissions, artisans might be granted use of one. 

Jitsumei 実名 True Name

The "government name" used in documents and registries. All classes would have a Jitsumei. 

Imina 諱 Real Name

Almost entirely interchangable with a Jitsumei.  However, it is also used in Buddhist contexts, as in it's the "real name" of a person as they would use when they enter heaven or hell, and can sometimes become presumptuous to use while they are living.

Azana 字 The Courtesy Name

In the style of Chinese scholars, this is the exact same concept as a "Courtesy Name" borrowed from China. Often, Monks and Scholars (people likely to have visited China to study or converse) would have an Azana, or Courtesy Name.  In short, instead of using one's jitsumei or nanori, you would use their azana as a courtesy for not overusing their real/true names, because that could be seen as overly familiar. If they had an azana, that would be used instead of a nanori. 

Alternative/Less Common Name Types

Japanese Christian Names

Houmyou 法名Buddhist Names

And Lay-Monkhood Notations 入道

Warawana 童名Childhood Names

Osanana 幼名 Infant Name

Gagou 雅号 Master Artist/Craftsperson Name

Songou 尊号 Honourific Name 

(Imperially granted, often in early history)

Potentially Presumptuous Name Types

Kabane/Sei 姓

Kabane are not really "names" but they are work/occupational titles included in your name to show social standing.  For example, it would be the name of the government or clan office you are assigned to, and then you would included a rank/position within that office.

And the rank within the Kabane you would normally have, called Kan'i 官位.  It would be noted with the possessive particle "no" and then the rank.  There are only a few rank descriptors, so you have:

• Ason, Asson, Asomi 朝臣

It's definitely fine to have a Kabane if you are LARPing as a high-ranked person or ranked highly in the SCA, in an attempt to incorporate that rank in your persona name.

Kaimyou 戒名 Posthumous Name

Ingou 院号 Palace Names

Shigou 諡号 Imperial Posthumous Names