Both首and頸are used for くび, which usually means "neck", but in Japanese 首 also means everything from the neck above. So yes, a secondary meaning of 首 is "head." For example, 首を垂れる means to hang your head down, not just your neck :). It old Japanese, 首 also means "face." The third kanji you noted is a simplified version of 頸, sometimes used when writing by hand but almost never in official print. The Japanese word that specifically refers to the head and not the neck is 頭 ／あたま。

くび does mean neck. The kanji is usually 首. Head is pronounced as あたま and the kanji is 頭.