The Massachusetts Open Data Initiative seems to provide only a link to the data, not the data itself. Their terms of use couldn't be really applied here.
The terms and conditions of the Massachusetts Archives website refer to the documents, not to their data collections.

As to historical vital records, [this page][1] states that "[t]he Archives' collections are public records and are open to all for research." Besides, it states

> Publication 

> The Archives requires patrons who copy materials for
> publication to complete a permission to publish form agreeing to use a
> standard citation for archival materials and to give a copy of the
> published work to the Archives. There are no charges for publication.

> Copyright 

> Records created by Massachusetts government are not
> copyrighted and are available for public use. Copyright for materials
> submitted to state agencies may be held by the person or organization
> that created the document. Patrons are responsible for clearing
> copyright on such materials.

There is also A Guide to Massachusetts Public Records Law on their website.

I'd suggest contacting archives staff to clarify the issues with a permission to publish form and a proper citation.

Hope it helps.

(edited in response to Rob's clarification)  

1. Sorry, I focused on this precise example and missed your more general question. Let's make clear some terms, shall we? For any data distribution process, there are the following participants:

- data collector (a person or persons who actually collected the data); 
- author/creator/owner/principal investigator - it might be either the collector herself or an organization, e.g. government agency, university, research institute, which the collector has collected the data for;
- data publisher (keeps the data and provides it for someone's use);
- data distributor (it might be the same organization as publisher, but also might be another organization, say, with broader opportunities to disseminate information)

    Now, the copyright always remains with the author/owner. And whatever other participants do, the data belongs to the principal investigator.  
Data provider/publisher shall comply with the owner's terms of use, especially regarding confidentiality.  
Data distributor shall comply with the data publisher's terms of use, hence, automatically, with the owner's.  
When someone looks for data, one should apply the terms of use attached to the data, whether they are written by the owner, publisher, or distributor.  
In this case, the data owner should be, in my understanding, the Massachusetts Registry of Vital Records and Statistics, the data publisher should be the Massachusetts Archives, and the Massachusetts Open Data Initiative goes as a distributor.  
Can you download the data from the MODI? No. There is only a link, so forget about their terms of use. Can you get the data from the archives? Yes. OK, then you should comply with their terms of use. Specifically: not with the terms of use of their website but with the terms of use of their data collections under the Massachusetts law.  
According to my experience, a possible citation for this data might look something like this: _Massachusetts Registry of Vital Records and Statistics, 2013, Vital Statistic Records (1841-1910). Boston, MA: Massachusetts Archives [distributor]_, though I'd hesitate to use it without contacting the archives.  

2. _What is the purpose of an Open Data directory if the collections it links to are not open data?_ Excellent question. Well, public records _are_ open data, are they not? The fact that you should do some paperwork for the data publisher doesn't make this data closed or private. But beside this, I totally agree with you: the ultimate goal of any open data distributor should be to provide open data, not to complicate the whole process.  

3. As discussed above, 2 out of 3 terms of use are not applicable here (one refers to a link, another to a website, but only the third one mentions the actual data).  
Have to say, I really sympathize with you about what you are going through. (The archives didn't answer even to phone calls? What's that again, the government shutdown?)
  [1]: http://www.sec.state.ma.us/arc/arcres/residx.htm