1. Functional - social/ economic
2. Hedonic - happiness, health and lifespan
3. Evolutionary - reproductive success
4. Religious - salvation and theosis
Psychology may be, broadly, defined as the science (or more accurately systematic-knowledge; Wissenschaft) that is related-to behaviour - including all forms of thinking (mental activity) as behaviours.
1. Functional - this sees psychology in terms of the perfomance of social functions, including economic activites; and normal everyday functions such as vision and hearing. The idea is that human cognition and behaviour are 'for' doing the things that people do, when they are being functional. This type of psychology is human orientated - interested my memory, learning etc - it was the original kind of psychology of Wilhelm Wundt's or Pavlov's lab work, and William James's Principle of Psychology textbook. When applied to animals, it regards the animals everyday-functioning as the subject - and in that sense is not really biological.
2. Hedonic - this sees psychology is a more humanistic' - way, as being about happiness, fulfillment, misery, alienation as immediate outcomes; and about living a long and healthy life as a long-term outcome. It is the psychology of Freud, Jung, Maslow, Rogers and Self-Help.
3. Evolutionary - this sees psychology as a branch of biology (man as an animal); hence with reproductive success and natural selection as the bottom-line. It focuses on selection pressures, adaptations, and has a timescale of generations.
4. Religious - this sees human psychology in terms of its interactions with outcomes such as salvation, theosis (becoming more divine), karma, reincarnation... whatever are the main outcomes in a particular religion; how to modify behaviour in pursuit of desired religioous goals (such as conversion, obedience, devoutness); and what kinds of being, thinking, and doing that affect these kinds of outcomes.
One can see that it makes a qualitative difference how psychology is defined in terms of its bottom-line; and there is no reason why we should expect one kind of psychology to map-onto, be integrable-with, another kind. They are doing different things.