TLDR: Eye blinking does occur in social contexts, but also reflexively (to moisten the eye) and in situations where the individual feels relaxed or is mentally processing. It's impossible to conclude any attempt at communication from eye blinks alone.
I've had a look at the study Eye-Blink Behaviors in 71 Species of Primates that you linked in your question. The scientists helpfully indicated the different families of primates in their diagrams and I noticed that in some diagrams the hominidae are all clustered together (indicating a strong correlation) and in others they are all over the place (indicating a weak or no correlation). Same with the other families and species.
I'm not a scientist, but my interpretation of these diagrams is that if blinking is a social interaction, then it's not a universal behavior consistent over different species, even those of the same order of species (= primates).
The scientists themselves conclude:
[T]he blink rate was not correlated with body weight but with the group size. This result is quite suggestive. From a physiological perspective, a rate of 2 bpm has been reported to provide sufficient corneal wetting for human adults , . Regardless of whether this basic rate is constant against body size factors, the observed blink rate was clearly higher than this rate in 65 of 66 diurnal primates. Our data suggest that these “additional” blinks may be related to social factors.
They also mention a correlation between blink rate and vigilance. Humans and primates blink less in stressful situations (to not miss any visual clues of danger). Since the primates in this study blink more in bigger groups, it indicates that blinking is a sign of feeling safe in bigger groups.
The same correlation is drawn in the study Eye Blink Rates and Eyelid Twitches as a Non-Invasive Measure of Stress in the Domestic Horse:
Eye blink rate has been used as an indicator of stress in humans and, due to its non-invasive nature, could be useful to measure stress in horses. Horses exhibit both full and half blinks as well as eyelid twitches. [...] Observation of eye blinks and eyelid twitches can provide important information on the stress level of horses with a decrease in eye blinks and an increase in eyelid twitches in stressful environments.
A study about more general facial expressions Human attention affects facial expressions in domestic dogs found that:
We presented dogs with an experimental situation in which a human demonstrator was either attending to them or turned away, and varied whether she presented food or not. Dogs produced significantly more facial movements when the human was attentive than when she was not. The food, however, as a non-social but arousing stimulus, did not affect the dogs’ behaviour. The current study is therefore evidence that dogs are sensitive to the human’s attentional state when producing facial expressions, suggesting that facial expressions are not just inflexible and involuntary displays of emotional states, but rather potentially active attempts to communicate with others.
Unfortunately I couldn't find any more studies about the eye blink rate in animals in a social context.
Apart from measuring stress, eye blink rate has been linked to cognitive processes, mental processing of events and learning.
The study Tracking Real-Time Changes in Working Memory Updating and Gating with the Event-Based Eye-Blink Rate states:
Effective working memory (WM) functioning depends on the gating process that regulates the balance between maintenance and updating of WM. The present study used the event-based eye-blink rate (ebEBR), which presumably reflects phasic striatal dopamine activity, to examine how the cognitive processes of gating and updating separately facilitate flexible updating of WM contents and the potential involvement of dopamine in these processes. [...] Together, these findings show that the ebEBR – an inexpensive, non-invasive, easy-to-use measure – can be used to track changes in WM demands during task performance and, hence, possibly striatal dopamine activity.
Or in easy English: When you need to switch from one active thought to a completely different thought, you'll probably blink.
The same conclusion is drawn in The Eye Blink as a Physiological Indicator of Cognitive Workload:
This report explores the hypothesis that eye blinks serve as a sort of “mental punctuation” during completion of a complex task. Specifically, it examines the rates of eye blinks while people are engaged in a complex decision-making task. The data suggest that eye blinks are suppressed while people are engaged in cognitive processing, and that they rebound once that processing has been completed.
The cow could have been absorbed with the puppy, then looked around and thought "oh, there's a human" blink.
The same thing might just have happened to you when you thought "I never knew that" blink.