The meaning of the term "aspirated"

Warning: I have no phonetic training, so this is just pieced together based on a few documents that I have read.

Phonetic aspiration

In IPA, the symbol /ʰ/ is used to mark aspiration, and this seems not to be a coincidence. Apparently, aspirated stops are followed by a noise resembling the sound /h/, in addition to the release noise of the stop (Kiss).

One common way of measuring aspiration is to look at the "voice onset time": the time between the release of a stop and the start of voicing. Aspiration is associated with a higher voice onset time, but there is a gradient between aspirated and unaspirated stops. Obviously, voice onset time can only be measured when the stop is followed by a voiced segment.

Daniel Harbour's answer to the ELU question "Can a plosive be pronounced without an audible release sound following it?" says:

(1) The stops t, p, k, when syllable-final, undergo glottal reinforcement in English. This minor glottal occlusion does not wholly impede the airstream. So, when the stop is released, the remainder of the air is released too. This is reminiscent of an unvoiced schwa, which accounts for what you hear after the t in cat. In terms of IPA transcription, one tends not to write the fine, automatic phonetic detail, and, so, for English, one marks only the preglottalization, as in [kʰæˀt].

(2) Crosslinguistically, the behaviour of syllable-final t, p, k varies. In Kiowa, for instance, as described by JP Harrington, t and p undergo complete glottal closure and are unreleased. In German, they are lightly post-aspirated. There are, however, only a certain number of perceptually distinct things you can do with your articulatory tract. So, my guess is, there'll be other languages that behave as English does.

As far as I can tell, word-final voiceless stops only seem to be described as phonetically "aspirated" when they are followed by an audible [h]-like sound that is sustained for a relatively long time. It doesn't seem to be standard to use the term "aspirated" to refer to all word-final voiceless stops that have audible release of any kind. I would guess that, as with stops in word-initial position, the classification would be gradient: stops that have no noticeable "puff of air" sound would be classified as having "no audible release" (also, perhaps misleadingly, called "unreleased"), stops with audible release that have a very small "puff" of air would be classified as unaspirated with audible release, and stops with audible release that have a "puff" or air of comparable duration to the VOT of a word-initial stop would be classifed as aspirated.

Phonological aspiration

When talking about English phonology, the term "aspirated" is usually only used to describe a set of allophones of the voiceless stop phonemes /p t k/ and the affricate /tʃ/. (Some other languages such as Hindi are sometimes said to have "aspirated voiced stops"; in that context, it refers to breathy-voiced stops).

I suppose, since English word-final voiced stop phonemes may be phonetically voiceless, they might be able to have some phonetic aspiration when they are audibly released, but I don't think that is all that common. No phonological description of English that I have read mentions any aspirated allophones of voiced stop phonemes.

The use of aspirated allophones in English

As the notes on that video say, "At the end of most words (and syllables) stop sounds are pronounced without a puff of air. For example, "lap," "sheep," "update." [...] if you make a puff of air, you’ll be understood, but you won’t be speaking with an American accent." There is no exception for words that end in consonant clusters. It is usual to not aspirate the stop.

In fact, I think a speaker would probably be less, not more likely to aspirate a word-final stop after /s/, because stops are typically not aspirated after fricatives in the syllable onset (e.g. star, spoon, skin), so I think English speakers have a general preference not to aspirate a stop that comes after a fricative in the same syllable. (See Kiss.)

I would guess words with a liquid followed by a word-final stop like "belt", "port", "part" are probably about as likely to be pronounced with aspiration as words with a vowel followed by a stop. Note that in American English accents, /r/ typically does not prevent a following /t/ from becoming voiced and flapped/tapped before a vowel-initial word--e.g the /t/ in "art" would be voiced and flapped/tapped in the context "the art of...".

The use of audible release in American English

I don't think it sounds bad to have a weak audible release for uttereance-final plosives in American English.

References

"Aspiration of stops after fricatives in English: Results from a pilot experiment", Zoltán G. Kiss