"Aspiration" is used in multiple ways, phonetically and phonologically, which can lead to some confusion, and Wiki reflects that confusion. Given that you're appealing to physical production and not phonology, I assume you don't care about aspiration as a phonological property, you just are about production (Wiki articles have many authors so there is no consistent perspective). Phonetically,
there literally are no voiced aspirates. Ladefoged & Maddieson proffer a phonetic definition
Having a greater rate of airflow than occurs in modal voice before or
after a stricture; arytenoid cartilages may be further apart than for
voiceless sounds
They also distinguish aspiration from breathy voice = murmur, which is
Vocal folds vibrating but without appreciable contact; arytenoid
cartilages further apart than in modal voice; higher rate of airflow
than in modal voice
You can see that "breathy voice" combines the properties of aspiration (increased airflow) with the properties of voicing (vocal fold vibration).
The Ladefoged and Maddieson definition defines a basically phonetic distinction contrast to a different type of phonation, namely modal voice.
A voiceless aspirated stop has a greater rate of airflow after its stricture, compared to a (modal) unaspirated voiceless stop. This does not mean that there is no airflow after release of a stricture – there always is airflow. What it means is that there is a high rate of airflow after the release of stricture, but all stops have some increased airflow after the stricture is released. Instead, we talk of the release burst of [t], vs [tʰ]. A voiced stop may also have a release burst, so you may detect a "puff of air", but it's not a big puff of air.
The tactile percept of a "puff of air" is not a particularly reliable means of measuring airflow - you need an airflow mask to measure airflow. The increased airflow associated with release of stricture is very high, very briefly, as the intraoral pressure is relieved and the vocal folds can move back into position for voicing. Aspiration is more than just that release, it is the entire pattern of increased airflow. Phoneticians usually speak of voice onset time as the measure of "aspiration", so aspiration is not a fundamental phonetic term, it's a phonological terms that phoneticians employ because of phonological contrasts, and they want to say something about the phonetics of [pʰ] vs [p] in Thai, English or Navaho.
The reason that people thought of "aspiration" in terms of a puff of air is historical (from old ideas about phonetics). The term "aspiration" was invented in the context of knowledge of Indo-European languages, and given the similarity between Sanskrit "bh" versus Greek "ph", it's not inappropriate to extract the unifying property, the "puff of air", but that's not how we define laryngeal states now.
Apart from the phonological unity of breathy voiced stops and voiceless stops with long VOT lag, there is an Indic-specific phonetic feature that encourages the use of the expression "voiced aspirate" (thus the Wiki article).
There is a difference in the production of the breathy stops in Indic versus breathy voiced stops found in a number of African language such as Shona. The difference resides in part in the type of breathy voicing (turbulent versus nonturbulent, respectively) and, apparently, the timing of the wide-glottis configuration (late, in Indic). One would not think of the typical non-Indic breathy voiced stops as having a "puff of air", but that is a typical and appropriate percept of how Indic [bʰ] et al. are produced.
