Is Mindfulness a poor translation for Sati? Usually Sati is translated as Mindfulness, but according to the Wikipedia entry, Mindfulness is described as follows: 


  Mindfulness is the psychological process of bringing one's attention to experiences occurring in the present moment, which can be developed through the practice of meditation and other training. The term "mindfulness" is a translation of the Pali term sati, which is a significant element of Buddhist traditions.


According to suttacentral's entry, Sati is defined as memory, recognition, consciousness, intentness of mind, wakefulness of mind etc.

My question is then, is it a mistake to call Sati Mindfulness, since Minsfulness means awareness. Now Sati is always used with his close companion sampajañña, which in turn makes the attention always directed to moral attitudes etc., whereas "mainstream mindfulness" just observes inner and outer phenomena with an open, curious and beginners mind.

Source: 
https://en.m.wikipedia.org/wiki/Mindfulness

https://suttacentral.net/search?query=sati
Wikipedia does not accord with the Pali suttas and the PTS Pali English Dictionary entry at Sutta Central is confusing, given it includes numerous different English terms. 

Pali words can generally only be understood in their contextual meaning. A sutta that clearly contextually describes 'sati' is MN 117, as follows: 


  One is mindful to abandon wrong view &amp; to enter &amp; remain in right view: This is one's right mindfulness.
  
  One is mindful to abandon wrong resolve &amp; to enter &amp; remain in right resolve: This is one's right mindfulness.
  
  One is mindful to abandon wrong speech &amp; to enter &amp; remain in right speech: This is one's right mindfulness.
  
  One is mindful to abandon wrong action &amp; to enter &amp; remain in right action: This is one's right mindfulness.
  
  One is mindful to abandon wrong livelihood &amp; to enter &amp; remain in right livelihood: This is one's right mindfulness.


This contextual meaning accords with a later Abhidhamma commentary, as follows: 


  What is sammasati? Sati means to bear in mind or bring to mind. Sati is the state of recollecting, the state of remembering, the state of
  non-fading, the state of non-forgetting. Sati means the sati that is a
  Spiritual Faculty, the sati that is a Spiritual Power, Sammasati, the
  Sati that is an Enlightenment Factor, that which is a Path Factor and
  that which is related to the Path. This is what is called sammasati."
  [Vbh.105, 286]


Ajahn Jayasaro properly explains 'sati' in his video called: Buddhist Meditation (4) Sati Sampajanna.

Similarly, Ajahn Buddhadasa properly explains 'sati' in his lecture called: The Scientific Cure of Spiritual Disease, as follows:


  Sati (mindfulness, reflective awareness, recollection) is the quick awareness and recall of the things which must be recalled. It must be
  as quick as an arrow. We also can describe sati as a vehicle or
  transport mechanism of the fastest kind. This most rapid transport
  doesn't carry material things, it carries wisdom and knowledge. Sati
  delivers paññä (wisdom) in time to meet our needs. Through the
  practice of mindfulness with breathing, sati is trained fully
  
  Sati realizes that something is happening and recalls the paññä relevant to that event. Sati immediately transports the necessary
  wisdom to that situation in time to deal with any possible problems.
  Mindfulness comes first. That wisdom applied to the experience is sampajanna. Delivered on time by sati, wisdom-in-action deals with the immediate situation. Then, in the very moment when sampajanna goes to work, the power and strength of samädhi gives force and energy to wisdom so that it can cut through the problem. To the degree that there is samädhi, to that degree wisdom-in-action will be able to solve the problem. Paññä acts as the warehouse of accumulated knowledge and insight which sati draws upon to deal with the sense experiences.


While the standard definition of 'samma sati' in the Noble Eightfold Path, the Satipatthana Sutta and the Anapanasati Sutta is not inherently wrong, it confuses students and scholars alike because the definition includes many different concepts. This definition is as follows: 


  And what, monks, is right mindfulness? (i) There is the case where a monk is a contemplator / observer of the body in &amp; of itself — ardent, clear-comprehending &amp;
  mindful — putting away greed &amp; distress with reference to the world.
  (ii) He is a contemplator / observer of  feelings in &amp; of themselves — ardent,
  clear-comprehending &amp; mindful — putting away greed &amp; distress with reference to the
  world. (iii) He is a contemplator / observer of the mind in &amp; of itself — ardent,
  clear-comprehending &amp; mindful — putting away greed &amp; distress with reference to the
  world. (iv) He is a contemplator / observer of Dhammas in &amp; of themselves
  — ardent, clear-comprehending &amp; mindful — putting away greed &amp; distress with
  reference to the world. This, monks, is called right mindfulness.


The word that gives rise to confusion is 'anupassī', which means to 'contemplate' or 'observe' or 'see'.  Students and scholars alike focus on this translation 'contemplate' and take it to be the meaning of 'sati'. However, if the definition of 'samma sati' is carefully examined, it is discerned the roles of 'sati' are: 


Remembering to put away greed &amp; distress with reference to the world. 
Remembering to maintain the mind in the activity of 'contemplation'. 


Therefore, to paraphrase the Wikipedia definition into a Buddhist context: 


  Buddhist mindfulness is the psychological process of remembering &amp; not forgetting to bring one's attention &amp; wisdom
  to experiences so those experiences occur without covetousness, distress &amp; other forms of craving &amp; defilement; which can be developed
  through the practice of meditation and other training. The term
  "Right Mindfulness" is a translation of the Pali term 'Samma Sati', which is a
  significant element of the Buddhist Noble Eightfold Path.