I have to say that I have never come across so much misleading, conflicting and incorrect info as I have with Edmund and Jane Batter! I can understand why you are confused.
Let me say that I have not found anything concrete but I am convinced that Jane was not the daughter of Edmund (1608). Some people think they were siblings but a few others have suggested cousins.
Here are some references which are of interest I believe and are hopefully of interest:
A. - I don't believe everything here !
THOMAS ANTRAM was born 1607 in Salisbury, Wiltshire, England., and died June 24, 1662 in Salem, Massachusette.. He married JANE BATTER May 24, 1630 in Salisbury, Wiltshire, England.. She was born 1609 in Salisbury, Wiltshire, England., and died 1674 in Salem, Essex, Co, Mass..
Notes for THOMAS ANTRAM:
Will of Thomas Antram : 1607 to 1679.
unto my dear wife Jane ( Batter) Antram the home we live in and the land it sits on. unto my sons John,James Antram ye rest of the lands in my name.
last wish that ye sons bring from England to America family and friends to start a new life. Jun 7th 1678.
Children of THOMAS ANTRAM and JANE BATTER are:
i. OBADIAH A ANTRAM, b. June 07, 1640, Salem, Essex, Co, Mass.; d. 1666, "Death at See" West Indies.; m. MARTHA BAKER, May 15, 1665, Salem, Essex, Co, Mass.; b. 1642, Salem, Essex, Co, Mass.; d. Unknown, ?.
ii. HANNAH or MARY ANTRAM, b. July 16, 1643, Salem, Essex, Co, Mass.; d. Unknown, ?; m. ISAAC BURNAP, 8 Nov 1658; b. 1639, Salem, Essex, Co, Mass.; d. Unknown, ?.
2. iii. JOHN ANTRAM, b. January 29, 1645/46, Salem, Mass; d. May 30, 1719, Northampton Twp, Burlington, NJ.
iv. ABIGAIL ANTRAM, b. August 24, 1648, Salem, Essex, Co, Mass.; d. Unknown, ?; m. JOHN PHELPS, Unknown; b. 1645, Salem, Essex, Co, Mass.; d. Unknown, ?.
3. v. JAMES ANTRAM, b. September 12, 1657, Salem, Essex, Co, Mass.; d. Unknown, Mansfield Twp, Burlington, N.J.
Jane Batter and husband Thomas Antram, came to Salem, Mass. in 1635 aboard the ship James. They were Pilgrims.
C. The folowing is an excerpt from the link below:
Thomas Antram weaver (2): This passenger was probably the Thomas Antrum baptized at St. Edmund, Salisbury, Wiltshire, on 31 December 1601, and he was certainly the man of that name who married in the same parish on 24 May 1630 Jane Batter, sister of Edmund Batter, who also sailed on the James. The couple's first known child, for whom there is no further record, was baptized at St. Edmund on 6 April 1634. Another four children were baptized at Salem between 1638 and 1646.
Please look at this link:
I did find the date (in my travels) that the 'James' left Southampton for America - I think it was April and it arrived in June 1635.
To summarize, I didn't find any real evidence that Edmund and Jane were siblings either which led me to think that they probably were cousins.