Gita 10.31
पवन: पवतामस्मि राम: शस्त्रभृतामहम् |
झषाणां मकरश्चास्मि स्रोतसामस्मि जाह्नवी ||
Here , mostly people say here ram is referred to as bhagwan shri Ramchandra.
But there is difference between shastradhari (who took upon weapon) and kshatriya. Shri Ram is born kshatriya. He never took upon weapon on purpose.
Also
Anand ramayan clearly says
रामेण सदृशो देव न भुतो न भविष्यति।
There was no god nor there will be any god who is equal to bhagwan shri Ramchandra.
Moreover in Brihad Ramayan
न राघव समोदेवः क्कापि ब्रह्माण्ड गोलके।
There was no god nor there will be any god who is equal to bhagwan shri Ramchandra.
So it's quite clear that here "ram " is not said to shri Ramchandra.
Let's see Mahabharata adi parv 2.3
त्रेताद्वापरयोः संधौ रामः शस्त्रभृतां वरः।
असकृत्पार्थिवं क्षेत्रं जघानामर्षचोदितः।।
Mahabharata adi parv 2.3
In the interval between the Treta and Dwapara Yugas, Rama (the son of Jamadagni) great among all who have borne arms, urged by impatience of wrongs, repeatedly smote the noble race of Kshatriyas.
Ram is actually said to parshuram ji.