I know this database is out there but the first issue is that the Nathaniel Potter born to this Martha and George Potter was born in 1626. He simply cannot be the man who was signing the Compact for Government in Portsmouth in 1638 as he would have been only 12! That means that any daughter this couple had was not an aunt to Nathaniel's two sons, Ichabod and Nathaniel.
Second, the source of this data is loaded with misinformation. It is not even a decent secondary source since most of the "records" were taken from people's undocumented family files and group sheets that the company lifted from online files. I made a mistake in entering data many years ago on one of my 3rd great grandmothers and somehow it ended up in there.
Paul Gifford, who is a superb researcher went to London and pulled all the records from this family. In some cases the so-called records never even existed and as I indicated, he found that the son Nathaniel was simply not of age to be signing a legal document in 1638, therefore he could not be "our" Nathaniel Potter. Paul is a historian from the University of Michigan. I've met him in person and I would put much more faith in his work than a poorly developed database put out by Ancestry. Paul posted quite a lot on the Potters at the Genforum web site. You can go to genforum.com/potter and do a search for "Gifford" and you will find his posts. He originally replied to Perry Streeter who put up the George and Martha data but there are several other posts as well.
Many of the New England Potters have been falsely connected to this George and Martha, but no actual records exist (as opposed to one of Ancestry's poor databases) that confirm any blood connection of this couple with Nathaniel Potter of Portsmouth or with Robert Potter of Warwick, They did have a son Robert but he died very young - long before the one in Warwick appeared.
Whether or not the Potter immigrants to RI were connected needs to be evaluated with DNA testing of patrilineal descendants of each family. There really are not any records that confirm a connection although I suspect there may have been one.
Also, whether or not the London couple had a daughter Martha is not even the issue. There's just no credible record that shows that any daughter named Martha belonging to them actually married Thomas Hazard. The source you provide says they married in RI in 1629. Unless they were Native Americans that cannot be. RI was not settled by Europeans until 1636. That alone calls the database into question. Ancestry does not care about accuracy, it cares about dollars. I had a copy of the US and International Marriage Records CD and I tossed it years ago because there was almost nothing that could be confirmed in the RI records. I have the ability to look at the originals because I live in RI but I'll be darned if there are any originals from 1629.
I am not trying to give you a hard time. It can be so easy to get lured by some of the stuff here at Ancestry.com. I have this image of future generations chasing their tails as they try to confirm all the nonsense from this site that unsuspecting people have put in their files.
I've even found errors in vital records, so the lesson is that everything needs to be confirmed against independent sources.