Bertha was Christ and Tina's first child and unfortunately I have not been able to locate either Bertha's or Bessie's (Tina's earlier child) birth records yet-so have not seen the Peterson reference; still looking/hoping. I also believe that Berg was most likely a farm name, or chosen for being in/near a mountain, etc. However I do feel that, by the traditions of the same patronymic system, her Norwegian surname probably started with the A. that she used as her middle initial when she married Christ. Both Christ and the three nephews who followed him over used O. as a middle initial, in this time period a practice often followed by immigrants Americanizing their name.
Christ married in Verdal, Norway to Mette, who in the US was known as Matilda Hegdahl and died 21 Jan 1891 in Minneapolis. Do you mean by your last note that Bertha's mother's maiden name is listed as Peterson? The correct Bertha? Just like Bertina Hegdahl, there was at least one other Bertha Hegdahl around at the time. If I remember, Bertha E. I recall that one of Bertha's children has on their birth certificate a version of Berth's middle name 'Kaspara', as her maiden name. Until receiving your note I had assumed that the use of the title 'pige'(Norwegian miss) could be taken as an indication that Tina was and had been unmarried until 1893, but now am starting to wonder if she had married a Peterson before she did Christ; perhaps in the US, and the pige was more of a courtesy or habit when entered by their pastor. Christ and Mette had two sons, one in Norway and John Sigurd, who we see in the US Census information, here. Counting the six children born and five still living which appears, if I remember, in both the 1900 and 1910 US Census, that leaves unaccounted for one other child of Tina's (the Mali or Malli I have seen on some trees does not belong in this family of Hegdahls). Can you help at all with some definite information on that?
I have also tried to keep track of the first name changes with Tina and seem to have much the same as yours. I did note however that Christ's death certificate lists Bentina Berg. Given that this entry was typed, the handwriting of the day, and the potential verbal confusion of Ber, Ben, and Bern spoken in Norwegian American to someone who was probably from somewhere else, I have been using Bertina, Berntina, Bentina, and even Albertina in searching. Have you found her in Norway?
I'll check the 1910 Census you refer me to-afraid to do it now and lose this note-and respond back to you later. Thank you for your responses!