The English language has incredibly many different regional accents, leading to the same words being pronounced differently by different people, sometimes in different places and other times in the same place.
What’s happening here is that some people say [ˈkʰlɪntən], such as Mrs Clinton herself, but others say [ˈkʰlɪ̃ʔn̩] with a glottal stop where the /t/ is phonemically. Those are not the only possibilities, either.
You cannot say that one accent is right and another wrong. They are simply different. It would be unnatural for someone whose accent requires one of those pronunciations to use the other one.
We aren't talking about some people pronouncing a word Fred and other people pronouncing that same word Wilma. Both [ˈkʰlɪntən] and [ˈkʰlɪ̃ʔn̩] use the same underlying abstract phonemes. Those two ways of saying Clinton are just alternate and perfectly valid phonetic realizations of the same phonemes as expressed in two different accents.
Phonemic /t/ has allophones of [tʰ], [t], [ɾ], and [ʔ], depending on a whole lot of factors.Native Native speakers will not hear a different word if you use a different allophone, because they won’t perceive a phonemic change just because an alternate allophone was used. It’s like did you coming out as didju when spoken rapidly.
Think of these three equivalent scenarios:
Imagine a Scottish speaker named Rory who for /r/ uses a flapped [ɾ] or a rolled [r]. You wouldn’t expect speakers of other accents to reproduce those allophones. It would still be the same word without the fancy /r/.
A name like Norbert or Bernard is always going to be pronounced differently by rhotic speakers as it is by non-rhotic speakers.
A name like Betty is always going to be pronounced differently by speakers who tap that /t/ as a [ɾ] compared with those who put a [t] there.
So both pronunciations of Clinton are perfectly normal ones in their respective dialects. The underlying abstract phonemes are the same; only the allophones are different, which means they are the same words.
Lastly, I would be very leery of expecting folks to reproduce an accent that is not their own. It will easily sound inauthentic, and may well be wrong. You just have to learn not to hear changes to allophones as being changes to the phonemes these represent. This can be especially hard for non-native speakers, because their brains work under different rules.