"Peter does not have hair on his head."
Translation:पीटर के सिर पर बाल नहीं है।
That is just a fact they overlooked. In Hindi "Hair" is almost always plural in sense: Your hair is red/he does not have hair.
That is the reason many Hindi speakers would reverse translate the sentence to "I have hairs on my head." And when they mean a single strand of hair, they would more likely say: I found a hair here. Because we think of them as a countable noun.
There is no way to literally translate the sentence 'Peter does not have hair on his head' into Hindi because Hindi does not have the verb 'to have'.
So, the sentence पीटर के सिर पर बाल नहीं है। literally meaning 'There is no hair on Peter's head' is used to communicate that.
Your sentence पीटर अपने सिर पर बाल नहीं है is grammatically incorrect because the verb is still conjugating with बाल while पीटर is left hanging.
You can say something like पीटर अपने सिर पर बाल नहीं रखता (Peter does not keep hair on his head) but the Hindi sentence, while grammatically sound, sounds just as awkward as its English counterpart. (This is the preferred form in some Hindi dialects, however. See WickeyMuis' comment below).
'on Peter's head' has the postposition पर at the end for the English preposition 'on'. The use of the postposition puts the preceeding word in oblique case. The fact that the word सिर is oblique is not actually visible, because with consonant endings in male singular nouns, direct and oblique case are the same. If you use the genitive postposition का / के / की (= English 's), it agrees with the form of the object that is possessed, i.e. 'head' in this sentence . Since सिर is masculin singular oblique, you need to use के as the respective form. Hope this helps.