1. I was thinking of an appliance, gives human the capability of ...
2. I was thinking of an appliance giving human the capability of ...
3. I was thinking of an appliance that gives human the capability of ...
Sentence #1 is grammatically incorrect. But both sentence #2 and sentence #3 are correct.
WHY sentence #1 IS INCORRECT, BUT sentence #3 IS CORRECT?
Consider the sentence below:
I was thinking of an [appliance] [(that) ___ gives human the capability of ...]
The part - that gives human the capability of ... - is a Relative clause and the antecedent of the relative clause is the Noun - appliance. The gap inside the relative clause is marked by the "___", is actually the subject position. The antecedent is clearly the subject of the relative clause. So whenever the gap of the relative clause denotes a subject position of the relative clause, the relative elements (e.g who, which etc) or subordinator (e.g that) is not omitted.
In your case the subordinator that is omitted, and therefore your sentence #1 is incorrect, but your sentence #3 is correct.
WHY sentence #2 IS CORRECT?
I was thinking of an appliance giving human the capability of ...
The predicator (FUNCTION), realized by the verb (WORD CLASS) - think, can license a Preposition Phrase (PP) as its complement - here, of an appliance giving human the capability of ....
Inside the PP structure there is a head Preposition - here, of - and the complement of the head Preposition is a Gerund-Participle clause - here, an appliance giving human the capability of .... In your case the Gerund-Participle clause has an explicit subject - an appliance.
(I didn't understand how omitting that just like your first sentence is relevant in the part after your "EDIT" in your question)
4. I saw a dead cat driving my car.
5. Driving my car, I saw a dead cat.
In these sentences the Gerund-Participle clause - driving my car - doesn't have an explicit subject. The subject is implicit - I. So naturally the structure of these sentence is not similar to that of the sentences in PART 1 of my answer. Therefore not comparable.
In case of sentence #2, the Gerund-Participle clause is a complement inside PP structure, that in turn is a complement of the verb - think. In sentence 4 and sentence #5, the Gerund-Participle clause is an adjunct. Hence, unlike sentence #2, the Gerund-Participle clause in sentence #4 and #5 are easily movable.