Does the Nihilism view (natthika-dithi) actually deny rebirths? Is the Nihilism view (natthika-dithi) actually denying rebirths?

In Fuller's The Notion of Diṭṭhi in Theravāda Buddhism: The Point of View, in the translation of the Sāmaññaphala-sutta, natthika-ditthi is formulated like this:


  Nothing is given, nothing offered, nothing sacrificed; there is no
  fruit or result of good and bad actions; no this world, no other
  world; no mother, no father; no beings who are reborn spontaneously;
  no good and virtuous recluses and brahmins in the world who have
  themselves realised by direct knowledge and declare this world and the
  other world.
  
  The person is composed of the four great elements; when
  he dies, earth returns and goes back to the element of earth, water
  returns and goes back to the element of water, fire returns and goes
  back to the element of fire, wind returns and goes back to the element
  of wind, while the senses disappear into space. [Four] men with the
  bier as a fifth take up the corpse, the funeral orations last as far
  as the burning ground, his bones are a dull white, his offerings end
  in ashes. They are fools who teach alms-giving. The doctrine of
  affirmation (atthika-vāda) is empty and false banter. Fools and wise
  alike are destroyed and perish at the breaking up of the body, they do
  not exist after death.


The following statements are related to 'rebirth':


Denying the "fruit or result of good and bad actions" -- which, except this-life fruit or results, are rebirth-related.
Denying "the other world" -- if it's not rebirth-related then it would be interesting to understand what "other world" is actually supposed to mean.
Saying "they do not exist after death" -- description of death as complete destruction (without rebirth).
Nothing is given, nothing offered, nothing sacrificed; there is no
  fruit or result of good and bad actions; no this world, no other
  worlds; no mother, no father; no beings who are spontaneously born;
  no good and virtuous recluses and brahmins in the world who have
  themselves realised by direct knowledge and declare this world and the
  other worlds.


I have amended the translation above. 


The word 'beings' ('satta')  is defined in the suttas as mental states of attachment &amp; identity (rather than physical life forms). 'Satta' is merely a 'self-view' held by Mara. 
The very fact that 'beings' are 'spontaneously' or 'immediately born' ('opapatika') due to acts of karma into the other worlds (heaven, hell, animal &amp; ghost) show it is not related to after-death-reincarnation or a new physical birth, as explained below:



  32. Sariputta, there are these four kinds of generation. What are the four? Egg-born generation, womb-born generation, moisture-born
  generation and spontaneous generation (opapatika). 
  
  Maha-sihanada Sutta



For example, the very moment the mind gets angry, it is spontaneously born as a 'hell-being'. 
'This world' ('loka') refers to the human world of good morals &amp; conscience. The 'other worlds' ('paraloka') refer to the heavenly (bliss), hell (torment), animal (immoral, emotionally reactive &amp; unreflective) and ghost (addiction) worlds. For example, MN 79 refers to the 'world of only pleasant feelings':



  Kiṃ panudāyi, atthi ekantasukho loko, atthi ākāravatī paṭipadā
  ekantasukhassa lokassa sacchikiriyāyā
  
  Udayi, is there a world of only pleasantness? Is there a course of
  actions to realise that world of only pleasantness?
  
  Here, Udayi, the bhikkhu secluded from sensual desires and thoughts of
  demerit abides in the first jhana: Overcoming thoughts and thought
  processs and the mind in one point internally appeased, without
  thoughts and thought processes abides in the second jhana. Again with
  equanimuity to joy and detachment, feeling pleasant with the body too,
  abides in the third jhana. To this the noble ones say abiding in
  pleasantness with equanimity. Udayi, this is the course of actions,
  for realising the world of only pleasant feelings (ekantasukhassa
  lokassa).









  The person is composed of the four great elements; when he dies, earth returns and goes back to the element of earth, water returns and
  goes back to the element of water, fire returns and goes back to the
  element of fire, wind returns and goes back to the element of wind,
  while the senses disappear into space. [Four] men with the bier as a
  fifth take up the corpse, the funeral orations last as far as the
  burning ground, his bones are a dull white, his offerings end in
  ashes. They are fools who teach alms-giving. The doctrine of
  affirmation (atthika-vāda) is empty and false banter. Fools and wise
  alike are destroyed and perish at the breaking up of the body, they do
  not exist after death.


This quote from the Samaññaphala Sutta is not a teaching of the Buddha but a teaching of Ajita Kesakambalin. In many places (eg. MN 115, MN 62, etc) it is taught life is only composed of elements (dhatu), which accords with Emptiness (sunnata). That is why the Buddha's response to the doctrine of Ajita Kesakambalin is similar but different, as follows: 


  'This body...is endowed with form, composed of the four primary elements, born from mother and father...subject to
  impermanence...dissolution and dispersion...this consciousness...is
  supported here and bound up here.'


There are two primary reasons why the doctrine of Ajita Kesakambalin (quoted) is 'nihilism'. 


The quote denies the efficacy (consequences) of kamma (actions). Thus, Buddhists who believe in present-life kamma-vipaka are not nihilists. 
The quote holds there is a 'person' or 'self' that ends at death. This is the precise definition of 'nihilism' found in the suttas. In other words, 'nihilism' is not the view of 'impermanence' but the view that a 'self' is impermanent. Thus, 'nihilism' is a wrong view because it is a 'self-view'. To quote: 



  Herein, a certain recluse or a brahmin asserts the following doctrine and view: ‘The self (attā), good sir, has material form; it is
  composed of the four primary elements and originates from father and
  mother. Since this self, good sir, is annihilated (ucchijjati) and
  destroyed with the breakup of the body and does not exist after death,
  at this point the self (attā) is completely annihilated
  (samucchinno).’ In this way some proclaim the annihilation (ucchedaṃ),
  destruction (vināsaṃ) and extermination of an existent being
  (sattassa).
  
  Brahmajāla Sutta


~~~


  And how, bhikkhus, do some hold back? Devas and humans enjoy being, delight in being, are satisfied with being. When Dhamma is taught to
  them for the cessation of being, their minds do not enter into it or
  acquire confidence in it or settle upon it or become resolved upon it.
  Thus, bhikkhus, do some hold back.
  
  How, bhikkhus, do some overreach? Now some are troubled, ashamed and disgusted by this very same being (bhaveneva) and they rejoice in (the
  idea of) non-being (vibhavaṃ), asserting: ‘In as much as this self
  (attaṃ), good sirs, when the body perishes at death, is annihilated
  (ucchijjati) and destroyed (vinassati) and does not exist (na hoti)
  after death (parammaraṇā) — this is peaceful, this is excellent, this
  is reality!’ Thus, bhikkhus, do some overreach.
  
  Iti 49


~~~


  What you, Kassapa, first called ‘suffering caused by oneself’ — this amounts to the eternalist theory (vadaṃ sassataṃ)….and ‘suffering
  caused by another’ — this amounts to the annihilationist theory (vadaṃ
  ucchedaṃ).
  
  Acela Sutta


~~~


  Don’t say that, friend Yamaka. Don’t misrepresent the Blessed One. It’s not good to misrepresent the Blessed One, for the Blessed One
  would not say, ‘A monk with no more effluents, on the break-up of the
  body, is annihilated (ucchijjati), perishes (vinassati) &amp; does not
  exist (na hoti) after death (parammaraṇā).’
  
  Then, friend Yamaka, how would you answer if you are thus asked: A monk, a worthy one, with no more mental effluents: what is he on the
  break-up of the body, after death?
  
  Thus asked, I would answer, ‘Form is impermanent… Feeling… Perception… Fabrications… Consciousness is impermanent. That which is
  impermanent is unsatisfactory. That which is unsatisfactory has ceased
  and gone to its end.’
  
  Very good, my friend Yamaka. Very good.
  
  Yamaka Sutta


In summary, often unenlightened people (puthujjana) falsely accuse Noble (Enlightened) People of being 'heretics' &amp; 'nihilists', as also occurred to the Lord Buddha himself: 


  37. "So teaching, so proclaiming, O monks, I have been baselessly, vainly, falsely and wrongly accused by some ascetics and brahmans: 'A
  nihilist is the ascetic Gotama; He teaches the annihilation, the
  destruction, the non-being of an existing individual.'
  
  "As I am not as I do not teach, so have I been baselessly, vainly, falsely and wrongly accused by some ascetics and brahmans thus: 'A
  nihilist is the ascetic Gotama; He teaches the annihilation, the
  destruction, the non-being of an existing individual.'
  
  "What I teach now as before, O monks, is suffering and the cessation of suffering.
  
  Alagaddupama Sutta: The Snake Simile