Pesahim 120b uses the phrase נים ולא נים תיר ולא תיר, 'dozing but not dozing, awake but not awake' to describe a minimum amount of halachic consciousness required to be considered not sleeping. The Talmud explains that in this state, a person responds, but cannot assert something independently. However, if reminded of something specifically, they can confirm or deny it.
The Rashbam explains that if asked where he put something, the dozer can't tell, but if he was asked if he put it in a specific place, he can say yes or no.
The Talmud gives an example through an anecdote. Abaye saw Rabah dozing while eating the final portion of matza. In order to make sure he was not a sleep, Abaye asked Rabah if he was just dozing, and Rabah confirmed that he was.
I found that this halacha applies at least to the following laws, which I am outlining based only on my rudimentary understanding of the Talmud:
- If a person falls asleep while eating the Pascal Offering they may not continue to eat it when they awake, (ibid).
- A person can not acquire a woman by way of Yibbum, or Levirate marriage, whilst asleep, (Yevamos 54a)
- A person may not eat in the hours before a fast day begins should they fall asleep, (Taanis 12a)
- A person fulfils their obligation if they doze whilst reading the Megillah, (Megillah 18b)
- A person has 'tasteless spittle', (one of seven substances that are applied to a stain to determine if it is blood or dye), if they sleep, or doze, at night, (Niddah 63a)
In any event, in order to not be considered 'asleep' in the above cases, a person must at least be able to speak a rudimentary response. The article you have cited says that a person experiencing sleep paralysis cannot speak.
It would seem that since they cannot speak, a person experiencing sleep paralysis could not respond to any sort of question, and would thus be considered asleep, according to halacha.1
1. As to the specific case you mentioned in your question, it is worth noting that even dozing is sleep-like enough for the case in niddah, resulting in 'tastless spittle'. Thus further investigation would be required to determine if dozing would count for shittin nishmei, though it seems that sleep paralysis would.