Is the first reaction really the "greater" or "lesser" jihad?
Yes, there is lesser/greater jihad.
Greater jihad/Major jihãd- One is fighting to overcome carnal desires and evil inclinations. It is known as the spiritual struggle, a struggle between two powers within ourselves: the soul and the body. The conscience is in conflict with the bodily desires. This spiritual conflict is an ongoing jihãd within each one of us. Islam expects its followers to give preference to the soul and the conscience over the body and its desires.
Lesser jihad/Minor jihãd - It is the armed struggle. However, that does not automatically mean unjustified use of violence. The minor jihãd may be divided into two: aggression and defense. Aggression against any people is not permitted in Islam; however, defense is an absolute right of every individual and nation. Islam has allowed the minor jihad only to defend the Muslim people and their land, and to maintain peace in Muslim societies.
The first verse of the minor jihãd, the armed struggle, revealed at that time is in Chapter 22, Surah Al-Hajj, of the Qur’ãn, verses 39-40. It clearly explains the purpose of the minor jihãd:
“Permission is granted to those who are fighting because they have been oppressed…those who have been expelled from their homes without any just cause…” (Surah al-Hajj, 22:39-40)
Again, referring to the non-believers of Mecca who waged war after war against the Prophet and his followers in Medina, the Qur’ãn in Chapter 2, Surah al-Baqara, verse 190, says:
“Fight in the way of God those who are fighting against you; and do not exceed (the limits). Verily Allãh does not love those who exceed (the limits).” (Surah al-Baqara, 2:190)
In this verse, the talk is about responding to a war by defending yourself; there is no talk of initiating aggression at all. Even in the defensive mode of struggle, Almighty God warns the Muslims that they should not “exceed” beyond the proper limits.
Islam teaches that Muslims should be strong in order to defend themselves, but that does not mean they have to become aggressive and unjust. In Chapter 8, Surah al-Anfal, verses 60-61 of the Qur’ãn, God has provided this general guidance very clearly when He addresses Muslims in the following way:
“Prepare against them (i.e., the enemy) with whatever force and trained horses you can in order to frighten thereby Allãh’s enemy, your enemy, and others besides them who you do not know but Allãh knows them.” (Surah al-Anfal, 8:60)
After giving this general guidance of being strong and prepared to defend ourselves, the verse goes on:
“But if they (the enemies) incline to peace, then you (also) incline to it, and put your trust in Allãh…” (Surah al-Anfal, 8:61)
In short, Islam wants Muslims to be strong so that others would not bully them; but then they have to extend the hand of peace even towards their enemies if there is an inclination of peace on the enemy’s part.
Actual Jihad definition
There are the defenition of jihad taken from various sources. More or less they mean the same but just for clarity i am posting them all.
Jihad (English pronunciation: /dʒɪˈhɑːd/), an Islamic term, is a religious duty of Muslims. In Arabic, the word jihād translates as a noun meaning "struggle". Jihad appears 41 times in the Quran and frequently in the idiomatic expression "striving in the way of God (al-jihad fi sabil Allah)"
Others use the term as a synonym for a struggle of any type. This reflects the origin of the word from the Arabic verb "jahada" which means to struggle or fight.
Jihad has the literal meaning of exerting our best and greatest effort to achieve something. It is not the equivalent of war, for which the Arabic word is qital. Jihad has a much wider connotation and embraces every kind of striving in God’s cause
the word jihãd literally means striving and working hard for something.
Below are wrong definitions. It is based on the mentality of western non-muslims
Many accounts in the media define "jihad" as a synonym for "holy war," -- a vicious clash between followers of different religions, each of whom believes that God is on their side and that the other side is is of Satan. This usage often appears on Western TV, radio, and other media during news about the Middle East, where it is used to describe a call for Muslims to fight against non-Muslims in the defense of Islam. Some Muslims have begun to adopt this meaning of "jihad" as a result of Western influence.
The word ‘Jihād’ is commonly translated into English as ‘the Holy War’
and for a long while now the word has been interpreted so that it has become
synonymous with a ‘mania of religion’. The word ‘Jihād’ conjures up the vision
of a marching band of religious fanatics with savage beards and fiery eyes
brandishing drawn swords and attacking the infidels wherever they meet them and
pressing them under the edge of the sword for the recital of Kalima. The Artists
have drawn this picture with masterly strokes and have inscribed these words
under it in bold letters:
The History of this Nation is a tale of Bloodshed.
The following are the sources from where i have picked up this content