The Templars absolutely were a Christian military order, and not a masonic one.

They were founded as an explicitly Christian order in the 1110's, and then formally approved by the Papacy in 1139. Brethren had to swear monastic vows of poverty, chastity, and obedience. They would also recite the paternoster a great many times each day and sing the canonical hours, all traditional features of Christian monastic life.

The Temple was also the first miltiary order, initially tasked with protecting pilgrims in the Holy Land, though this soon extended to military service in defence of the Crusader States as well. The Hospitallers, another military order, were founded before the Templars, around the 1070s, but did not militarise until the mid-twelfth-century. The Temple's internal structure reflects this military purpose. The brethren were divided into three classes, knights, sergeants, and chaplains. Only the chaplain class did not participate in warfare. The sergeants were a mixture of administrative and military servants, whilst the knights were the Order's main military component and also most of its leadership. The Temple did not run hospitals or focus much on charitable works, like the Hospitallers and Teutonic Knights did. Instead, their military campaigns were their prime focus.

The main controversy around the Order was their arrest and trial on charges of heresy from 1307-12, but criticism of the Templars, and military orders in general, had been around since their inception in the early twelfth century. Initially many monastics struggled to accept the concept of a military order, arguing that a military role was incompatible with monastic life. However, there were some major ecclesiastics that did endorse the orders. Aside from the Papacy, which repeatedly patronised the military orders and established new ones, Bernard of Clairvaux was a leading supporter of the Templars in their early years. Criticism increased after the loss of Acre in 1291, the last Latin stronghold in the mainland Levant, with the orders being blamed for this defeat or criticised for not retaking the Holy Land after it. The trial of Templars ended with the Order's suppression in 1312, the Pope judging that though the charges were probably false, the Temple's reputation had been too tainted to continue.

The masonic 'link' is one created in the early modern period when Freemasonry, like many other groups, was trying to create a longer and more dramatic history for itself. There is no evidence of a link between the two groups whatsoever.

So there isn't any controversy over whether they were a Christian or military order, and there certainly isn't over any supposed links to Freemasonry. The only real debatable point I could suggest would be to look into whether they and other military orders were really monastic or not. Some historians consider them monastics, other see them as lay religious, or as their own special group between the two.

Sources:

Malcolm Barber, The New Knighthood (Cambridge, 1994).

Helen Nicholson, Templars, Hospitallers, and Teutonic Knights : images of the military orders, 1128-1291 (Leicester, 1993).

Helen Nicholson, The Knights Templar (Sutton, 2001).

John Walker, ''The Templars and Everywhere': An Examination of the Myths behind Templar Survival after 1307', in Jochen Burgtorf, Paul F. Crawford, Helen Nicholson (eds), The Debate on the Trial of the Templars (1307-1314) (Abingdon, 2010), pp. 347-357.