Yes, you do have that option to register as an independent voter, semantics and terminology aside. (Independent is a term a nationwide audience would understand best, right?) Any registered voter can vote in the general election. Otherwise, correct.
The board of registration is a few blocks from me. I need to drop by and pick up some forms. Lots of work to do...
In TX, one is automatically considered an "unaffiliated voter" at registration. He registers by party (for two years, no more, no less) by voting in a primary election. Primaries are held the second Tuesday of March in every even-year. If some offices do not produce majority-backed nominees in the primary, there is a second or runoff primary on the second Tuesday of April. But if one misses either of those primaries, he is an unaffiliated voter for two years -- until the cycle repeats itself, 2006, 2008, etc. Of course, an unaffiliated voter can call himelf a member of a major party, but he is technically not a major party member in TX unless he votes in the every-other-year primaries.