In philosophy, there are many ways of waging war. One way is violence, another is sparing intellectually. I believe that at its core, Jihad is about purifying the self and using that vessel to help purify others. You don't need an external enemy to struggle, you can struggle against your own imperfections. You don't have to kill when raiding, merely offer a different point of view that leads to another contemplating it. If you are strong physically, you can do missionary work rather than conquest. There are other ways to experience a Jihad other than violence. While I'm not Muslim, I've always respected it for that purpose, and making alms giving a duty.
This is your feelings on the matter... But since I am not feeling merciful I will give references.
"Not equal are those who sit and recieve no hurt, and those who strive and fight in the cause of Allah with their goods and persons. Allah hath granted a place higher who strive and fight with their goods and persons than to those who sit. Unto all Allah hath granted good: But to those who strive and fight Hath He distinguished above those who sit, by a special reward." Quran 4:95
-The Jihad al-Akbar (greater Jihad) is to fight and strive against Kufr, not Jihad al-Nafs or Jihad al-Shaytaan which is to sit and strive against ones self or the entity and concept of Shaytaan.
"And there is life for you in the retaliation, O men of understanding, guard yourselves (from the Kufr or Munafiqeen)." Quran 2:179
- this has a meaning at a micro level, that one must defend one's body from enemy or from Kufr, as to retaliate and destroy the influence, this the life is in the destruction of such Fitnah. However, apply this to the macro level, which is always to be done (if not then you break Tawheed al-Haakimiyyah and Tawheed al-Rabuubiyyah), it points to the destruction of Fitnah is a benefit to the Muslim and the whole of the Earth. Thus one must attack the Fitnah through all means necessary in every part of the world for simply existing it is a danger to your Imaan and Taqwa. Then comes the war against all for the sake of removing Fitnah in all cases and without exception for to leave any without a treaty is to allow Fitnah knowingly, further it is clear that all outside of belief is Fitnah unless it is under rule of the Muslim and thus his Hadood is above them satisfying Rabuubiyyah and his Shariah is placed on the believers which satisfies his Haakimiyyah.
"Fighting is prescribed for you, and ye dislike it. But it is possible for ye to dislike that which is good for you (or gives benefit, Maslahah), and ye dislike that which is bad for you. But Allah knoweth and ye does not." Quran 2:216
- even if it is adverse to you, the physical jihad is prescribed, as it was for the Salaf in the times of Rasul.
"Let those fight in the way of Allah who sell this life (Dunya) for the other (Jannah). Whoso fighteth in the cause of Allah, be he slain or victorious, on him shall we bestow a great reward." Quran:4:74
- to give one's life here and all its worried for the cause to increase one's Imaan and Taqwa warrants a great award, beyond Jannah. To be one who sacrifices, Fedayi, is the cause for the rewards of Allah.
"The punishment for those who wage war against Allah and His messenger and strive to cause Fitnah in the land is this, that they should be killed or crucified or their hands and feet should be cut on opposite sides or they should be imprisoned; this shall be as a disgrace for them as on this earth, and in the afterlife there shall be grievous chastisement." Quran 5:33
- those who wage war upon Allah, hmm, what does this mean when applied to Rabuubiyyah (Allah is the lord of all)? The Kufr or disbelief which creates Fitnah (mischief in the land), is in constant war against Allah and his Lordship. Thus to utterly eliminate is the goal and commandment of this Surah.
"And fight with them until there is no more Fitnah, and religion is all for Allah." Quran 8:39
- fight all Fitnah until there is none. Very clear, this is a clear cut issue and all scholars of the past agree on this issue.
"O ye who believe! When ye meet those who disbelieve in battle, turn not your backs to them, unless maneuvering in battle or intent to join another company, he truly has incurred the wrath of Allah, and his habitat will be in hellfire, a haples journey." Quran 8:15
- to shy away from the battle except as posturing or planning incurs wrath. To retreat in the face of Fitnah is Kufr and this is clear.
"Go forth light armed and heavy armed, strive with your possessions and lives in the way of Allah! That is best for you, if ye but knew." Quran 9:41
- go no matter your power, light or heavy. This can mean the Jihad al-Nafs, but the verb usage to go emphasized one who loves and takes to an action as opposed to Nafs. Thus it can only be Harb/Ghazw, etc or Dawah.
"Allah hath purchased of their persons and goods; for theirs is the garden: they fight in his cause and slay and are slain: A promise binding on Him in truth, through the Law, Gospel and the Quran: and who is more faithful in his covenant than Allah? then ye rejoice in the bargain ye have concluded: an achievement supreme." Quran 9:111
-to lose all things and fight is the achievement supreme. One cannot lose possessions in Nafs.
"And when We wish to destroy a town, We send Our commandment unto the people of it who lead easy lives, but I they transgress therein; thus the word proves true against it, so We destroy it with utter annihilation." Quran 17:16
- the transgression of the enemy is always Kufr and thus the prescription is Harb, hence Dar al-Harb = Dar al-Kufr. This is clear cut.
This is only some from the Quran. I will not delve into Hadith or official Fatwa or Hukm, as it is exceedingly time consuming. But the Jihad is more than what you say...
Also you say purify others, what do you believe that means? I am sure after reading, you are assured.