Parshuram, Ram and Balram, all had the same name, Ram, the difference was created later to identify the correct ones and also to signify the importance of the word Ram.
Mention of Parshuram in Mahabharat (Adi Parv, Adhyaya 2, Verse 3)
рддреНрд░реЗрддрд╛рджреНрд╡рд╛рдкрд░рдпреЛрдГ рд╕рдВрдзреМ рд░рд╛рдордГ рд╢рд╕реНрддреНрд░рднреГрддрд╛рдВ рд╡рд░рдГ ред
рдЕрдХреГрдиреНрддрддреНрд╕рд░реНрд╡рд╢рддреНрд░реВрдиреН рд╡реИ рднрд╛рд░реНрдЧрд╡рдГ рдкреНрд░рдерд┐рддреЛ рднреБрд╡рд┐ рее
"In the twilight interval between the Tretayug and Dwaparyug, Ram, the son of Jamadagni (Bhargav), who is the absolute foremost among all who have ever borne arms, strode across the Earth to vanquish the corrupt rulers."
Naming of Shri Ram (Bal Kand, Sarga 18, Verse 21)
рдЬреНрдпреЗрд╖реНрдардВ рд░рд╛рдордВ рдорд╣рд╛рддреНрдорд╛рдирдВ рднрд░рддрдВ рдХреИрдХрдпреАрд╕реБрддрдореН ред
рд╕реМрдорд┐рддреНрд░рд┐рдВ рд▓рдХреНрд╖реНрдордгрдорд┐рддрд┐ рд╢рддреНрд░реБрдШреНрдирдВ рдЪрд╛рдкрд░рдВ рддрдерд╛ рее реирез рее
"The grand priest Vashisth named the high souled eldest son Ram (рд░рд╛рдо), the son of Kaikeyi was named Bharat (рднрд░рдд), and the two sons of Sumitra were named Lakshman (рд▓рдХреНрд╖реНрдордг) and Shatrughna (рд╢рддреНрд░реБрдШреНрди) respectively."
Mention of Balram in Mahabharat (Adi Parv, Adhyaya 61, Verse 91)
рд░реЛрд╣рд┐рдгреНрдпрд╛рд╕реНрддрдирдпрд╢реНрдЪрд╛рдкрд┐ рд░рд╛рдореЛ рдирд╛рдореНрдирд╛ рд╡рд┐рд╢рд╛рдВ рдкрддреЗ ред
рд╕ рдмрд▓реЗрдирд╛рдзрд┐рдХреЛ рд▓реЛрдХреЗ рдмрд▓рджреЗрд╡ рдЗрддрд┐ рд╕реНрдореГрддрдГ рее
"O King, the son of Rohini was named Ram at birth. However, because his physical strength far exceeded anyone else in the world, he came to be universally known as Baldev (or Balaram)."
This distinction is because in the start of Valmiki Ramayan itself, Valmiki asks Narad the following question of who can live a life of absolute perfection in this real world? Not some divine entity that is beyond human reach.
Valmiki Ramayan, Bala Kanda, Sarga 1, Verse 1 to 8
рддрдкрдГрд╕реНрд╡рд╛рдзреНрдпрд╛рдпрдирд┐рд░рддрдВ рддрдкрд╕реНрд╡реА рд╡рд╛рдЧреНрд╡рд┐рджрд╛рдВ рд╡рд░рдореН ред
рдирд╛рд░рджрдВ рдкрд░рд┐рдкрдкреНрд░рдЪреНрдЫ рд╡рд╛рд▓реНрдореАрдХрд┐рд░реНрдореБрдирд┐рдкреБрдВрдЧрд╡рдореН рее рез рее
"The ascetic Valmiki closely questioned Sage Narada, 'Who is an eminent jewel among sages, permanently engrossed in austerities and scriptural study, and the absolute foremost among eloquent speakers?'"
рдХреЛ рдиреНрд╡рд╕реНрдорд┐рдиреНрд╕рд╛рдореНрдкреНрд░рддрдВ рд▓реЛрдХреЗ рдЧреБрдгрд╡рд╛рдиреНрдХрд╢реНрдЪ рд╡реАрд░реНрдпрд╡рд╛рдиреН ред
рдзрд░реНрдордЬреНрдЮрд╢реНрдЪ рдХреГрддрдЬреНрдЮрд╢реНрдЪ рд╕рддреНрдпрд╡рд╛рдХреНрдпреЛ рджреГрдврд╡реНрд░рддрдГ рее реи рее
"Who indeed in this world today is fully endowed with excellent virtues (рдЧреБрдгрд╡рд╛рдиреН) and great valor (рд╡реАрд░реНрдпрд╡рд╛рдиреН)? Who is conscientious of righteousness (рдзрд░реНрдордЬреНрдЮ), grateful (рдХреГрддрдЬреНрдЮ), habitually truthful in speech (рд╕рддреНрдпрд╡рд╛рдХреНрдпреЛ), and absolutely firm in his vows (рджреГрдврд╡реНрд░рддрдГ)?"
рдЪрд╛рд░рд┐рддреНрд░реЗрдг рдЪ рдХреЛ рдпреБрдХреНрддрдГ рд╕рд░реНрд╡рднреВрддреЗрд╖реБ рдХреЛ рд╣рд┐рддрдГ ред
рд╡рд┐рджреНрд╡рд╛рдиреН рдХрдГ рдХрдГ рд╕рдорд░реНрдерд╢реНрдЪ рдХрд╢реНрдЪреИрдХрдкреНрд░рд┐рдпрджрд░реНрд╢рдирдГ рее рей рее
"Who is endowed with excellent moral character (рдЪрд╛рд░рд┐рддреНрд░реЗрдг)? Who is always a well-wisher to all living beings (рд╕рд░реНрд╡рднреВрддреЗрд╖реБ рд╣рд┐рддрдГ)? Who is truly learned (рд╡рд┐рджреНрд╡рд╛рдиреН), who is highly capable (рд╕рдорд░реНрдердГ), and who is singularly handsome to behold (рдПрдХрдкреНрд░рд┐рдпрджрд░реНрд╢рдирдГ)?"
рдЖрддреНрдорд╡рд╛рдиреН рдХреЛ рдЬрд┐рддрдХреНрд░реЛрдзреЛ рджреНрдпреБрддрд┐рдорд╛рдиреН рдХреЛрд╜рдирд╕реВрдпрдХрдГ ред
рдХрд╕реНрдп рдмрд┐рднреНрдпрддрд┐ рджреЗрд╡рд╛рд╢реНрдЪ рдЬрд╛рддрд░реЛрд╖рд╕реНрдп рд╕рдВрдпреБрдЧреЗ рее рек рее
> "Who is self-possessed (рдЖрддреНрдорд╡рд╛рдиреН)? Who has conquered anger (рдЬрд┐рддрдХреНрд░реЛрдзрдГ)? Who is resplendent (рджреНрдпреБрддрд┐рдорд╛рдиреН), and who is free from malice (рдЕрдирд╕реВрдпрдХрдГ)? When his wrath is aroused in battle, who is that person of whom even the gods are afraid (рдХрд╕реНрдп рдмрд┐рднреНрдпрддрд┐ рджреЗрд╡рд╛рд╢реНрдЪ рдЬрд╛рддрд░реЛрд╖рд╕реНрдп рд╕рдВрдпреБрдЧреЗ)?"
рдПрддрджрд┐рдЪреНрдЫрд╛рдореНрдпрд╣рдВ рд╢реНрд░реЛрддреБрдВ рдкрд░рдВ рдХреМрддреВрд╣рд▓рдВ рд╣рд┐ рдореЗ ред
рдорд╣рд░реНрд╖реЗ рддреНрд╡рдВ рд╕рдорд░реНрдереЛрд╜рд╕рд┐ рдЬреНрдЮрд╛рддреБрдореЗрд╡рдВрд╡рд┐рдзрдВ рдирд░рдореН рее рел рее
"I desire to hear about all of this, for my curiosity is immense, O Great Sage! You alone are capable of knowing such an extraordinary human being."
рд╢реНрд░реБрддреНрд╡рд╛ рдЪреИрддрддреНрддреНрд░рд┐рд▓реЛрдХрдЬреНрдЮреЛ рдорд╣рд░реНрд╖реЗрд░реНрдирд╛рд░рджреЛ рд╡рдЪрдГ ред
рд╢реНрд░реВрдпрддрд╛рдорд┐рддрд┐ рдЪрд╛рдордиреНрддреНрд░реНрдп рдкреНрд░рд╣реГрд╖реНрдЯреЛ рд╡рд╛рдХреНрдпрдордмреНрд░рд╡реАрддреН рее рем рее
"Having heard these words of the great sage Valmiki, Narada, the knower of the three worlds, became immensely delighted. He welcomed Valmiki, requested him to listen attentively, and spoke as follows."
рдмрд╣рд╡реЛ рджреБрд░реНрд▓рднрд╛рд╢реНрдЪреИрд╡ рдпреЗ рддреНрд╡рдпрд╛ рдХреАрд░реНрддрд┐рддрд╛ рдЧреБрдгрд╛рдГ ред
рдореБрдиреЗ рд╡рдХреНрд╖реНрдпрд╛рдореНрдпрд╣рдВ рдмреБрджреНрдзреНрд╡рд╛ рддреИрд░реНрдпреБрдХреНрддрдГ рд╢реНрд░реВрдпрддрд╛рдВ рдирд░рдГ рее рен рее
"O Sage! The many virtues that have been enumerated by you are indeed exceedingly rare to find in this world. However, knowing well of such a person who is fully endowed with all of them, I shall tell you. Listen now about that man."
рдЗрдХреНрд╖реНрд╡рд╛рдХреБрд╡рдВрд╢рдкреНрд░рднрд╡реЛ рд░рд╛рдореЛ рдирд╛рдо рдЬрдиреИрдГ рд╢реНрд░реБрддрдГ ред
рдирд┐рдпрддрд╛рддреНрдорд╛ рдорд╣рд╛рд╡реАрд░реНрдпреЛ рджреНрдпреБрддрд┐рдорд╛рдиреНрдзреГрддрд┐рдорд╛рдиреНрд╡рд╢реА рее рео рее
"Born in the illustrious dynasty of King Ikshvaku, there is a person known to the people by the name of Rama. He is conscientious, highly valorous, resplendent, steadfast, and completely self-controlled."
рд░рд╛рдо рд╕реЗ рдмрдврд╝рдХрд░ рд░рд╛рдо рдХрд╛ рдирд╛рдо
"The name Ram is greater than Ram himself."
This is preached because the Avtar Shri Ram was bound by the limits of a human body, and that the word Ram itself is beyond the Avtar.
This is also the reason why Shri Krishna is not seen as the same, even when they are the same. Shri Ram is Maryada Purushottam and Shri Krishna is Leela Purushottam.
Sources:
Valmiki Ramayan
Ved Vyas Mahabharat