In the examples you give, the first has a British accent, and the second has an American accent
Britons and Americans are both lazy about the letter "t" but in different ways
Britons tend to reduce the "t" to a glottal stop so wanted becomes wan'id. The British speaker is pronouncing the "t", but it is reduced.
Americans tend to flap the "t" before an unstressed syllable. It is this flap sound that is confusing you.
She'd always wanted to go to Thailand
In this recording, the speaker is pronouncing the T, but is kind of reduced, as JamesK's answer says.
He wanted that job so badly he was willing to kill for it.
In this one, there's neither a glottal stop, nor a flap (as the other answers suggest). It's a case where Americans usually drop the T entirely.
In most--if not all--American accents, when a stressed syllable ends in an N and the next unstressed syllable starts with a T, the T is usually dropped, so ˈin.ter.net becomes innernet, ˈcoun.ter becomes counner, ˈtwen.ty becomes twenni, ˈwan.ted becomes wannid etc., so you're hearing it corrrectly: "/ˈwɒnɪd/".
I believe this is called "glottalizing", where one syllable in a word is weaker than the other and the leading "t" in the weaker is almost dropped. Other examples would be where an American says "waw-der" for "water", or where a Scottish person says "Sco'ish". More details here.
It has no effect on meaning.
