The claim
"The Islamic concept of Jihad refers exclusively to violent holy war against non-Muslims."
The short answer
The Arabic word 'jihad' literally means 'struggle' or 'striving.' It includes the daily struggle against one's own ego and desires. Armed jihad is only one category, and it is regulated: the Prophet ﷺ forbade the killing of women and children, and peace is to be accepted when the enemy inclines to it.
Evidence-Based Clarification
Jihad Includes the Struggle Against the Self
The Quran uses jihad for striving in Allah's cause in every form, and the Prophet ﷺ called the one who strives against his own soul a mujahid. The often-quoted report about returning 'from the lesser jihad to the greater jihad' is weak and should not be used as proof, but the point it makes is supported by authentic narrations. Armed jihad is one category, and it is bound by rules: the Prophet ﷺ forbade the killing of women and children, religion may not be compelled, and peace is to be accepted when it is offered.
PRIMARY SOURCES (4)
"And strive for Allah with the striving due to Him."
وَجَاهِدُوا فِي اللَّهِ حَقَّ جِهَادِهِ
— Surah Al-Hajj 22:78check the original
"The Prophet ﷺ said: 'The mujahid is the one who strives against his own soul.'"
— Jami' at-Tirmidhi 1621check the original
al-Tirmidhi: Hasan sahih · al-Albani: Sahih
"A woman was found killed in one of the battles of the Prophet ﷺ, so the Messenger of Allah ﷺ forbade the killing of women and children."
— Sahih al-Bukhari 3015check the original
al-Bukhari: Sahih
"And if they incline to peace, then incline to it also and rely upon Allah."
وَإِن جَنَحُوا لِلسَّلْمِ فَاجْنَحْ لَهَا وَتَوَكَّلْ عَلَى اللَّهِ
— Surah Al-Anfal 8:61check the original