New York State Census of Inmates in Almshouses and Poorhouses

Hendrickson, the reason you’re in a quandary is that you are trying to cite that image as though you consulted an original record collection at the NY State Archives. But you didn’t.

• Basic Fact 1: You’re using an image provided by a database at Ancestry, an image within a block of cards that Ancestry has taken out of context and segmented into its own organizational scheme.
• Basic Fact 2: The manner in which that set of cards is imaged does not enable you to create a full citation from what you, yourself, see there. If you were in archive, you would not only have the card, but you'd have the box in which the card is filed, with Box No., Collection Name/No., Series Name/No., and Record Group Name/No.  Or you'd have the microfilm with a label that provides ID details for a citation and you'd have the background information that almost always appears at the start of microfilmed records.
• Basic Fact 3: Because of Fact 2, you cannot possibly create a full and reliable citation to the original document in Layer 1.  Either your citation will be incomplete or else you will "borrow" details from Ancestry's own citation, trusting that those details are correct. As you've discovered, parts of it may or may not be correct. That's why the "citing" information from a website provider goes in it's on layer. After we've identified the provider of the image, we then say what that provider tells us about the source.

Commendably, you’ve gone to the NYSA website to study its catalog to find out more about this set of records. The discrepancy you report between Ancestry’s source note and NYSA's cataloging details is a cautionary tale for all researchers. It may be that the person who created that source note at Ancestry made an error. Everyone makes errors. It may also be that the NYSA has multiple microfilm rolls with similar content and that Ancestry filmed one while we've found the cataloging details for the other. When I’ve made a point of tracking down the cause of such discrepancies between archives cataloging and the source notes of online image providers, I’ve found cases of both.

All things considered, it’s wise to follow the most basic rule of citation: We cite what we use.  

Yes, most researchers want the lead words of a citation to reflect the identity of the document. In this case, the title of Ancestry’s database does that.

EE would create a two-layer citation, with the first layer citing the database (and the item it delivers), while the second layer quotes Ancestry's source note:

“New York, U.S., Census of Inmates in Almshouses and Poorhouses, 1830–1920,” database with images, Ancestry (https://www.ancestry.com/imageviewer/collections/1083/images/31864_243994-00050 : accessed 8 October 2022) > Suffolk > 1902, imaged card for Alfred Davis, Record No. 816; citing “New York State Board of Charities, Census of Inmates in Almshouses and Poorhouses, 1835–1921, Series A1978, microfilm, 225 rolls.”

Incidentally, the wording I’ve quoted in the “citing” layer comes from the “Original data” section of Ancestry’s “Source Information” block at the main page for this database (https://www.ancestry.com/search/collections/1083/).