Did Ichigo originally have Sode no Shiraiyuki? At the start of the Bleach manga, Rukia stabs Ichigo in the heart with her zanpakuto in order to give him her shinigami powers, which allows him to protect his family from the Hollows.

After that, Ichigo obtains a sealed zanpakuto that he uses (alongside Rukia in a gigai) to fight Hollows around Karakura. 

Later, Rukia returns to Seireitei and regains her shinigami powers, which robs Ichigo of his.

This implies that Ichigo had borrowed his powers from Rukia for the run of the Karakura Arc. This is further supported when Ichigo gains his own zanpakuto, Zangetsu, and his sword is unsealed rather than remaining in its more benign state.

Because Ichigo is borrowing Rukia's spirit powers, and because Zangetsu's form appears to be more volatile and less prone to remaining sealed, is the first zanpakuto that Ichigo carries Rukia's zanpakuto Sode no Shiraiyuki?
Ichigo doesn't have Sode no Shiraiyuki originally. He loses his powers originally because Byakuya severs his soul chain, which detaches him from being able to touch his powers. He regains the powers because of training with Urahara that also causes him to become 


  part Hollow.


As quoted from bleach.wiki.com:


  [Urahara] explains that Byakuya destroyed the source of his power, leaving him without any Shinigami powers and that he must restore these in order to fight the Shinigami. He explains about Reiryoku, saying that the more one's reiryoku rises, the sharper the movements of their spiritual body becomes. He says that if Ichigo can manage to move even better with his spiritual body than he can with his mortal body, then we will have made a complete recovery of his reiryoku. 


More importantly, however, his unsealed zanpakuto is the same both before and after his fight with Renji/Byakuya. To have had Sode no Shiraiyuki previously, he would have needed to have an entirely different sword, which he didn't.

These aren't exactly manga scans, but here are pictures of the sword before and after Ichigo regains his powers:

Before:



After: