What is morality according to Buddhism?

Conceptions of what is good and evil are depend on what frame of reference you apply. Certain religious traditions that sprang up in the Middle East, the earliest of which was Zoroastrianism which most probably influenced all that followed, see life as a battle between the fundamental and external forces of good and evil:

Zoroastrianism states that active participation in life through good deeds is necessary to ensure happiness and to keep chaos at bay. This active participation is a central element in Zoroaster's concept of free will, and Zoroastrianism rejects all forms of monasticism. Ahura Mazda will ultimately prevail over the evil Angra Mainyu or Ahriman, at which point the universe will undergo a cosmic renovation and time will end. Wikipedia

In Buddhism there is no cosmic fight between good and evil. Buddha pragmatically lays the origin of good and evil at the door of each persons individual actions. Greedy, hateful and delusional actions lead to bad results 3 Unwholesome Roots AN 3.69; Generous, compassionate and wise actions lead to good results - for you and the people around you.

In Buddhism, there is no fight between good and evil—that is the most wonderful thing in the Buddhist practice! There is no fight between good and evil. Good and evil are both organic matters. If you have understanding and wisdom, you will know how to handle both the flower and the garbage in you, you can make the Buddha be born every moment of your life, and peace and happiness will be possible. Thich Nhat Hanh

Why some people think "this same X is bad" and others think "this same X is good"?

They are applying their frame of reference to the world. For Buddha even if they think X is good but this belief leads to suffering then they are delusional.

An interesting and related topic worth mentioning here is the role of Mara. In Buddhist stories, that have their origins in the Pali Canon, relate how Buddha was tempted by Mara (a kind of demon figure) - Maara Legends in Canonical Texts. You could equate the temptation of Buddha by Mara to the temptation of Christ by Satan. The two stories are very similar. However, without the underpinnings of a cosmic battle between good and evil, Buddhism's view of what Mara is, or represents, in these stories is much more nuanced than Christianities concept of the devil. In Buddhism the general consensus is that Mara is merely the personification of all the unwholesome (akasula) qualities and so ...

Mara is not very pleasant, but if you know how to help Mara, to transform Mara, Mara will become Buddha. If you don’t know how to take care of the Buddha, Buddha will become Mara. Thich Nhat Hanh

i think Ambalatthika-rahulovada Sutta is a good sutta for this question.

from my study of Buddhism, what is good and bad can be determine by: 1) determine the intention. 2) Results. 3) compare it with the wise (is it blameless?). Tons of sutta about who is wise and who isn't.

The respective key words for "good" and "bad" in the suttas are "wholesome" (kusala) and "unwholesome" (akusala):

• kusala (adj.) [cp. Sk. kuśala] 1. (adj.) clever, skilful, expert; good, right, meritorious.
• akusala : (nt.) demerit; sin; bad action. (adj.), unskilful.

"Wholesome" and "unwholesome" here are qualifications for action/kamma. Keeping in mind that, to the Buddha, volition is kamma (see AN 6.63) manifested by body, speech and mind.

“And what, friends, is the unwholesome, what is the root of the unwholesome, what is the wholesome, what is the root of the wholesome? Killing living beings is unwholesome; taking what is not given is unwholesome; misconduct in sensual pleasures is unwholesome; false speech is unwholesome; malicious speech is unwholesome; harsh speech is unwholesome; gossip is unwholesome; covetousness is unwholesome; ill will is unwholesome; wrong view is unwholesome. This is called the unwholesome.

“And what is the root of the unwholesome? Greed is a root of the unwholesome; hate is a root of the unwholesome; delusion is a root of the unwholesome. This is called the root of the unwholesome.

“And what is the wholesome? Abstention from killing living beings is wholesome; abstention from taking what is not given is wholesome; abstention from misconduct in sensual pleasures is wholesome; abstention from false speech is wholesome; abstention from malicious speech is wholesome; abstention from harsh speech is wholesome; abstention from gossip is wholesome; uncovetousness is wholesome; non-ill will is wholesome; right view is wholesome. This is called the wholesome.

“And what is the root of the wholesome? Non-greed is a root of the wholesome; non-hate is a root of the wholesome; non-delusion is a root of the wholesome. This is called the root of the wholesome.

-- Sammādiṭṭhi Sutta, MN 9

(the causes are also repeated and further explained in the Nidana sutta (AN 3.33)).

The above definitions are more general, based on the 3 defilements (greed/lobha, hate/dosa, delusion/moha) but the list of mental states can expand to more specific ones. For example, any of the five hindrances are also understood as unwholesome. In Theravada tradition, the Abhidhamma and Visuddhimagga lists 10 defilements.

In another exposition, the Buddha further distinguishes between four kinds of kamma (see AN 4.235):

"Monks, these four types of kamma have been directly realized, verified, & made known by me. Which four? There is kamma that is dark with dark result. There is kamma that is bright with bright result. There is kamma that is dark & bright with dark & bright result. There is kamma that is neither dark nor bright with neither dark nor bright result, leading to the ending of kamma.

Thus, generally speaking, it is said that wholesome actions are those that lead to a favorable rebirth and lead to nibbāna, whereas unwholesome actions are those that lead to suffering, perdition, hell.

For an interesting read in this subject, see The Roots of Good and Evil by Nyanaponika Thera.