"Voiced" in the phonetic sense [1], i.e., spoken with the vocal cords vibrating. Voiced 'x' sounds like the /gz/ in "eggs", /ɛgz/; voiceless 'x' sounds like the /ks/ in the American English pronunciation of 'x' itself, /ɛks/.

Many, if not all, American English dialects pronounce the words in the fashion I describe. In them, the name 'Alexander' would be

    /ˌæ.lɛˈ(gz)æn.dər/

while 'Alexa' would be

    /əˈlɛ.(ks)ə/

- in both of which, the phoneme corresponding to the letter 'x' is parenthesized.

Generally in English 'x' is voiced when it precedes a stressed vowel, which it does in 'Alexander'; in 'Alexa', 'x' precedes a reduced vowel, and therefore would always take the unvoiced pronunciation. (It'd sound very odd to an anglophone ear otherwise - say /əˈlɛ.gzə/ one time out loud and see if you don't feel the same.)

That said, it wouldn't be incorrect to pronounce 'Alexander' in American English with an unvoiced 'x', as

    /ˌæ.lɛˈksæn.dər/

but, while I believe some dialects of English may default to this pronunciation, certainly not all do. (Neither of the dialects I speak does so, at the very least.) This pronunciation also produces a "hitch" or break in the word between the unvoiced 'x' and its preceding vowel, which would tend to make it a little odd both to hear and to say.

[1] https://en.wikipedia.org/wiki/Voice_(phonetics)

> I put the stress differently, but otherwise I would pronounce Alexa as Alexander cut short.

In “Alexander”, the normal pronunciation of the letter “x” is the voiced consonant cluster /gz/, in “Alexa”, it's usually the unvoiced cluster /ks/. (Also, the second “a” is usually different between the two, being /æ/, like the “a” in “pad”, in “Alexander” and /ɑ:/, like the “a” in “father”, in “Alexa”.)

The difference in the “x” is the normal way that the pronunciation of “x” differs when following a stressed vs. unstressed vowel in English.