Does that mean that the ancestor was born at the meetinghouse?
The short answer is, the person's birth date appears in the monthly meeting records. It could be as part of a family group record when the family joined that monthly meeting, and the birth may not have been in the geographical area of the meeting or even in Ohio. If the family has moved from other areas, there might be a notice about the family leaving in the previous meeting's minutes.
Before trying to answer this question, I think it is important to back up and do a bit of analysis of the work which prompted this question. You say that some of the places which are using the notation Stillwater MM, Belmont, Ohio are authored works (for a definition, see this answer) citing The Encyclopedia of Quaker Genealogy, Vol IV, By William Wade Hinshaw.
My guess (and this is only a guess) is that people have not actually looked at the minutes of the monthly meeting, or at Hinshaw's Encyclopedia, but are instead relying on a source which has abstracted Hinshaw's work, or Hinshaw's index volume, Index to Encyclopedia of American Quaker Genealogy. Looking at Google Books for Quaker histories, I see references describing people as being "of" a particular meeting. Just like other genealogical writings which say that someone is "from" a particular place, that doesn't mean the locality is that person's birthplace -- they could live in the area, but have actually been born in a different town (perhaps nearby).
That disclaimer aside, let's look at what format Hinshaw's Encylopedia uses. Since Volume IV is not available to view at Hathi Trust, I chose a page from Volume I:
You can see that the name of the Monthly Meeting is in the header at the top of the page, and that some, but not all, of the data abstracted in the Encylopedia has a birthplace noted. My guess is that the people consulting Hinshaw have substituted the name of the Monthly Meeting for the birthplace because that is the only place information they have, in the same manner that someone researching in the UK might list the GRO Registration District that the birth was registered in as a placeholder, when they don't have the birth certificate in hand.
To see what information is actually listed in the Meeting minutes, you would have to try to find the original records. Ancestry.com has a downloadable Research Guide to Findng Your Quaker Ancestors which shows what the originals might look like. The guide says:
Monthly meetings often kept birth registers. Births are sometimes
listed by date and sometimes by family group. If the births were
recorded at a later date, registers may name each child in the family
in succession.
Along with names of children, birth dates, residence, and parents’
names, registers sometimes include details about marriages or deaths
of children listed. You may also find details on the mother’s
parents.
The Family History Library Catalog lists this transcription and index for a yearly Meeting held at Stillwater:
Transcription and index to the minutes of Ohio Yearly Meeting of Friends held at Stillwater, P. O. Barnesville, Ohio, 1907
To look for other records, try the suggestions in the Family Search Research Wiki article on US Quaker Research. I do not see the Stillwater Monthly Meeting in the collection U.S., Quaker Meeting Records, 1681-1935 on Ancestry.com (disclaimer: I haven't searched the entire catalog).
Resources:
Update: The Quaker research guide at AmericanAncestors.org (NEHGS) has the link to Monthly Meetings in North America: A Quaker Index - quakermeetings.com, an online version of Monthly Meetings in North America: An Index by Thomas C. Hill.
Searching for Stillwater yields two different meetings by that name, each with their own information page describing what records are extant and where they are held.
STILLWATER (FGC) OHIO GUERNSEY & BELMONT QUAKER CITY 43773, 24001 LEATHERWOOD RD. (S.R. 265), AT MILLWOOD TWP.RD. 9435, S.14, T.9, RGE.__
STILLWATER (WILBURITE) OHIO BELMONT BARNESVILLE 43713, WARREN TWP., 61830 SANDY RIDGE RD.
Click on the names of each meeting to see the history of each meeting, where the records are held, and other information like local histories which may have references that will lead you to other records.
