A major reason for calling Iblees (Satan or Shaitan) a disbeliever in this verse is that he disobeyed (an order of) Allah.
As the opposite of obedience is disobedience. And the opposite of Islam is kufr (disbelieve). Islam إسْلام also means submission (~obedience).
(34. And (remember) when We said to the angels: "Prostrate yourselves before Adam.'' And they prostrated except Iblis (Shaytan), he refused and was proud and was one of the disbelievers (disobedient to Allah).) (From tafsir ibn Kathir)
An other reason also explained by ibn Kathir in his tafsir is arrogance and we could also add rebellion against a clear order of Allah:
"Iblis, the enemy of Allah, envied Adam because Allah honored Adam. He said, `I was created from fire, and he was created from clay.' Therefore, the first error ever committed was arrogance, for the enemy of Allah was too arrogant to prostrate before Adam.'' I - Ibn Kathir - say, the following is recorded in the Sahih,
«لَا يَدْخُلُ الْجَنَّةَ مَنْ كَانَ فِي قَلْبِهِ مِثْقَالُ حَبَّةٍ مِنْ خَرْدَلٍ مِنْ كِبْر»
(No person who has the weight of a mustard seed of arrogance in his heart shall enter Paradise.)
Iblis had disbelief, arrogance, and rebellion, all of which caused him to be expelled from the holy presence of Allah, and His mercy.
Ibn Kathir also quoted that Iblees was imitating the angels in their worships and tried to be even "better" than them in worship and knowledge. He already knew about the living on earth, as he was among the residents of earth. That's why he was also called to prostrate before Adam beside the Angels (so ibn Kathir considered him a Jinn whom pretended to worship Allah in the best manners).
Many earlier tafsirs like at-Tabari and al-Qurtobi even say that Iblees was an Angel before this quoting statements of sahaba like ibn 'Abbas and ibn Mas'ud and tabi'yn like ibn Jurayj, ibn al-Mussayib and Qatadah.
قال ابن عباس : وكان اسمه عزازيل وكان من أشراف الملائكة وكان من الأجنحة الأربعة ثم أبلس بعد (al-Qurtobi)
Ibn 'Abbas said: his name was 'Azazeel عزازيل and he was among the noblest of the angels and had four wings than he became desperate أَبْلَسَ (ablas ~ Iblees)
In an other quote ibn 'Abbas said that after his rebellion or disobedience Allah became angry and cursed him and he became a shaitan. In an other statement of ibn 'abbas his name in Syriac was 'Azazeel and in Arabic al-Harith الحارث and he was one of the guardiance of Janah.
Ibn 'Achour in his tafsir at-tahriri wa-tanwir discusses the meaning of:
وكان من الكافرين
which in your quited translation doesn't show the real time of the verb as كان is a past tense so the more correct translation would be:
he was among the disbelievers
as it is in my quote from ibn Kathir!
Saying that scholars accepted that Iblees was a Kafir even before the order of worship, but they differ on the meaning, as before this act disbelieve didn't exist, some also say he was a disbeliever before the revelation of the verse, some said he was a kafir in the knowledge of Allah (that means Allah knew he was only acting if he was the most active worshiper, but in truth he was a disbeliever), some also said Iblees was showing worship and obedience while hiding disobedience (he was a hypocrite). Ibn 'Achour also pointed at the emphasize of the word استكبر which means more than simple arrogance but it means a great arrogance!
Most scholars say that the reason why he was considered a disbeliever are: his disobedience of an order of Allah, his arrogance and rebellion against a clear order (while the angels only exclaimed their doubts about the creation of humans, as they saw what jinn did on earth, but accepted it because Allah is wiser and knew why he created humans). But also Iblees was a Jinn and we know that among them there are believers and disbelievers so it could be that he was a hypocrite or disbeliever who just showed that he was an active worshiper.