The Youtube video was broadly correct. Your daughter may have a slightly unusual accent for an American, or her perception of the sounds may be influenced by the spelling.
I say "broadly correct" because for most American English speakers, the sound used in a word like "little" is actually not exactly a [t] (voiceless alveolar plosive) or a [d] (voiced alveolar plosive). Instead, it would usually be a voiced alveolar "flap" or "tap"; this is represented as [ɾ] in the International Phonetic Alphabet. That said, native American English speakers generally aren't consciously aware of the difference between [d] and [ɾ] in English words; to many people, they both sound like /d/. (Both [d] and [ɾ] are voiced sounds, unlike [t]).
The symbol [ɾ] is also used to represent the sound of the letter "r" in Spanish, which is quite similar, but there may be some phonetic differences between these sounds (see this blog post by John Wells: tap, tap). It's important to keep your pronunciation of flapped American English "t" and "d" distinct from your pronunciation of the American English "r" sound, which can be written in various ways in IPA (such as /r/, /ɹ/, /ɻ/; the last is the most accurate of these transcriptions for cross-language comparisons). So for the purposes of foreign language learning, I'd say it's often an acceptable simplification to tell learners to just pronounce this sound as a quick [d].
Both /t/ or /d/ are normally replaced by voiced flaps [ɾ] in certain situations in American English, and in general, people can't hear a difference between the [ɾ] that replaces /t/ and the [ɾ] that replaces /d/. We say that /t/ and /d/ are "neutralized" in this context. For example, native American English speakers will generally hear both "atom" and "Adam" as [æɾm̩], and they may confuse these two words if there is not enough context. (This is not just a hypothetical example: this actually happened to me when watching the movie Real Steel, which has a robot character named "Atom" who I thought was named "Adam.") Here's another blog post from Wells that discusses this neutralization, and how some speakers may not realize that it happens in their accent: I don't believe it!
A lot more has been written about t- and d-flapping; the Wikipedia article is a good start. Another useful resource is this "Flap t FAQ" by Tomasz P. Szynalski. You can find more details by Googling something like "American English t flapping."
I don't think it has much, if anything, to do with speed. I would normally pronounce little as [lɪɾl̩] even in slow, careful speech. The only time I can think of when I might use [lɪtl̩] is when singing or reciting poetry. I also have not heard of any differences between the East Coast and the West Coast on this point of pronunciation: as far as I know, flapped /t/ exists in all American accents (although on the individual level, people may not always use a flap in all the words where it is possible).
That said, you will not be misunderstood if you use [t].