In the introduction in Book of Five Rings Musashi address himself as 新免武蔵守・藤原玄信 (Shinmen Musashi no Kami - Fujiwara no Harunobu), and throughout the book uses Shinmen Musashi. Shinmen is his father's clan name. I'm going to guess Takezō is either a different pronunciation of 武蔵 or a mistake on whatever translation you're reading. It would appear he changed his clan name to Miyamoto, but I don't know why.

Tokugawa Ieyasu you can find here

For big important samurai (sry, that rules out Musashi):

Think of clan names as companies. If you started out as a no-name commoner (百姓 hyakushō, literally "hundred surnames") like Hideyoshi then you better make up one. As samurai clans expanded (became big corporations), many were divided into branch-lineages (subsidiaries) - heck most samurai clans were supposedly branches of the Taira and Minamoto too (Musashi was supposedly Fujiwara, as was the supposed original Tokugawa). Some of these came to be identified by the branch name rather than the main name. A clan name change was usually for sociopolitical-economic ties, legitimacy, and/or prestige. These could be revival of main lineages like the Ise and Matsudaira into the Hōjō and Tokugawa respectively (or faked it). Or it could be an actual adoption into another clan like Nagao Kagetora getting adopted into the Uesugi clan or Hideyoshi getting adopted into first the Fujiwara then the Toyotomi clan.

Some positions were often restricted to certain clans, and even certain lineages of certain clans, as the Kantō Kanrei was reserved for the main Uesugi line, the Yamanouchi. Nagao Kagetora's adoption into the Yamanouchi Uesugi essentially merged the main Nagao and Uesugi lines. It gained him the position of Kantō Kanrei, and with it the claim and legitimacy over the Kantō region while the Uesugi an important ally and an heir (so like a corporate merger?). Ieyasu needed the dead Tokugawa name for recognition as Shugo of Mikawa. The main Nitta branch died out in the mid 14th century, and two surviving branches decided to fight for the name in the early Sengoku, with the Iwamatsu Nitta winning, henceforth being recognized as Nitta. Both the Hōjō and the Mori (and many others actually, it was common) also have younger sons adopted into and named as heirs to important "vassal" families, to essentially secure them.

The formal personal name (ie, Ieyasu) usually compose of two characters, one of which is often something usually used by members of the family. The other I am not really sure, though there are cases of it being "given" by a superior at genppuku (declaration of majority). Both could be changed on "adoption" into another clan, usually taking one character from the adopter. Or for Ieyasu, seems like just because he wanted to take it from some saying. I mean the Tokugawa branch (if he didn't make it up) was dead anyway, and he needed to give up Moto. You can also give one of your character to a vassal (without adoption or genppuku) to signify your ties.

One last note. Etiquette dictates you don't address someone by their formal personal name. You address them by their honorific title or if you are close enough to them their informal names. Musashi comes from the honorific title Musashi no Kami (Lord of Musashi). People in the Sengoku (at least) had a habit of styling themselves certain honorific titles without any formal recognition, which was most likely the case in this case.

Source: David Spafford. "What’s in a Name? House Revival, Adoption, and the Bounds of Family in Late Medieval Japan." Harvard Journal of Asiatic Studies. Volume 74, Number 2. December 2014.

EDIT: Fixed link.