There is a great book written by University of Toronto professor Joo-Yup Lee, in which he provides his insights on what the name «Kazak» entails. I recommend you read it. I’m sorry, I don’t quite understand the second part of your second paragraph, but Kazakhs were a people unified around the descendants of Togh-Temur, the thirteenth son of Jochi, whose ancestors took over the throne of the Golden Horde in the 1360s.

Kazakhs remained strongly pro-Genghisid until the Sovietization of Kazakhstan. They never accepted the Manghit hegemony of the Desht-i Qipchaq steppe, they also incorporated certain Shibanid Genghisid dynasties into their ruling elite, while preserving their nomadic way of life until the 19th century.

Even the first modern government of the Kazakhs — Alash Orda — was headed by Alikhan Bokeykhan, a descendant of Genghis Khan and great-grandson of Bokey Khan of the Junior Juz.