Indeed, those are interesting questions and difficult ones to answer - since reality is then much more complicated than it is displayed in the graph. First of all, we need to define what empathy is - and as an extra, second to that we need to differentiate the influencing factors that in combination add up to a certain personality, and we need to define the terms involved in this.
I. empathy // (there are various, yet similar approaches to this, the following is one of them)
empathy is a complex ability, that involves many areas of the brain. empathy is a sort of emitter / receiver system for interhuman interaction and mind to mind connection, and a tool of self and other imagination. yet, it says little about how nice someone is, how many feelings he has - empathy as such is simply a functional tool of cognition with neutral value. for example, it depends on the individual - wether autistic, neurotypical, psychotic or psychopathic or..
a. how functional, experienced and well trained his empathy tool is
b. how his current state of mind, his overall personality and his state of body affects the funtionality of the empathy tool
d. how and for what purpose he uses his empathy tool
e. how well or to which degree he can willingly control the empathy tool, to which degree he is subject to its automated, inuitive and instinct functioning
f. how and which empathetic information is transferred to others and received from others by the empathy tool
g. how other components of the brain and mind (i.e. amygdala for emotion) and their functionality affect the empathy tool and empathetic information
for empathy we need the cognitive ability of imagination, or of imagining something that is not - that is: not a concrete item or real object. necessary for this are well trained and fully functional executive cognition functions - and the deficit in these, due to low dopamine from birth onwards, is where autism/ADHD comes in. cognitive empathy is the cognitive ability of imagination used to imagine the others's and one's own mind - a sort of intellectual approach, also: perspective taking.
on the other hand, emotional empathy - although intertwined with cognitive empahty in reality - is recognizing and feeling in oneself the emotion expressed by the other (that is being affected by emotion of others) and feeling and expressing appropiate and relational emotion towards the other. cognitive empathy is a necessary factor for functional emotional empathy. if there is an autistic deficit of cognitive empathy due to underdeveloped executive cognition functions this may result in problems with emotion regulation and recognition in oneself (inability to name one's own feelings/emotions) and problems in self awareness and self imaging - as well as other awareness and recognition of emotions in others. plus, also in problems in emotional-social intuition and in automated receiving of, feeling of and responding to emotions of others.
finally, compassionate empathy is feeling compassion, which is sepparated from empathy and a social-moral function for social behaviour and for feeling socially-emotionally pressed to provide help to each other.
in conclusion: there is a difference a. if one has the functional ability of the empathy tool, but does not use it to feel much for others or has no interest in others - the reasons for this may be personality, very often personality disorder, depression, trauma, bad experiences, bad upbringing, being tired or bored, being drunk or high and many more, such as being an a**hole or not wanting to feel to much of others (because it is to annoying to take part in other's feelings) - and b. if one has a dysfunction of the empathy tool, but would like to use it to have and feel a better social and emotional connection to others, but cannot or cannot do so well simply because the brain has underdeveloped functionality, or damage, or illness.
II. brain, psyche, personality, character // the brain is not equal to the psyche. the psyche, however, equals the self. the self in a broader sense is the sum of an existance .> that is a person consisting of its physiological body and the body's informational processes (the mind, the spirit, the soul etc.), the latter being within the whole of the body but coordinated and centralized in the brain. the field of neurology focuses on the physiological, physical and biochemical structure and functioning of the brain as an organic body.
in a narrower sense the psychological self - or psyche - is the sum of the information and the informational processes (or meaningful movements, eg. thinking words (thinking = movement, words = meanings)) that are contained in the brain. therefore, this sum includes the stored and stationary information (information = meaningful associated data / meaningful things) and the processes retrieving, using, producing and storing this information, as well as all computations carried out with information from internal storage and external sources. the self / psyche is the informational-virtual body, to which the brain is the physical body or the physical basis used to store and compute the informational-virtual body / psyche.
therefore, the field of psychology focuses on the informational-virtual body, which in other terms and approaches may be referred to as self, psyche, spirit, soul, mind etc. or referred i.e. in german as "ghost" (geist), which underlines the seemingly non-physical nature of this thing in contrast to the physical body that produces it. there is no sharp line between psyche and brain - rather both influence each other, and as the human being is a self forming and self composing thing, the psyche can change itsself and it can also to small degree even change the physical body of the brain - for example by cognition training, or by resolving trauma. this is known as neuroplasticity. in terms of autism as a disorder or natural psycho-neurological type it is both a question of the psyche and of the brain, and thus all transmitters used in the brain - such as dopamine, which is the main connector for everything - are of interest for making (if necessary) improvements: like improving executive functioning, and thus: empathy.
personality then is a decriptive image of the structure of traits, that individually define the informational-virtual body - of course in combination with its underlying physical body, the brain. character is the set of moral, emotional, social and self and other related values incorporated into the personality.
autistic brains, psyches, personalities and characters are equally complex and diverse as so called neurotypical ones - although the latter can't imagine what whole problem is - and it is therefore for all always an important question to think, how their thinking influences their sense of being and their relation to others.
III. psychopathy, antisocial disorder (DSM), dissocial PS-disorder (ICD)... maybe tomorrow