When you are looking for a person in a record set, you'll have the most success when you know three things:
- the time the event took place
- the place the event took place
- the name (including variants) the person may have been listed by
The previous answer has already addressed the problem of the historical place name.
Research guides like the FamilySearch Research Wiki can help you find records that might contain the information you're seeking. Start with one of the main articles like Pennsylvania, United States Genealogy to get an overview, then look on that page for record type articles like Pennsylvania Vital Records and research strategy pages like How to Find Pennsylvania Birth Records. Long-distance researchers will also want to take note of the blue button that takes you to the summary of Pennsylvania Online Genealogy Records.
The Wiki says:
During the period 1855 to 1892, no births were recorded by the county
or state. You must search substitute records to locate your ancestor’s
birth date and place.
The article gives specific suggestions for collections at FamilySearch that may be useful, followed by general suggestions for records that might have birth information, with links to the article for each record type.
Try this collection first: Pennsylvania, Historical Society of Pennsylvania, Births and Baptisms - FamilySearch Historical Records
Then look for birth information that might be contained in other record sets.
Records that give birth information:
- Death Records
- Cemetery Records
- Church Records
- Military Records
You've already used some of these record types. To look for more records at FamilySearch, do a place search in the FamilySearch catalog. Check all jurisdictions that apply to the area, not just Warren County. Look at nationwide records, state-level records, county-level records, and records at the town level.
Search all relevant jurisdictions when searching for reacords on any site, when searching for finding aids and research guides, when using directory sites like Cyndi's List and Linkpendium, and when searching for published works on the Internet Archive, Google Books, HathiTrust, etc.
Tips for Newspaper research
When you're searching for birth notices in historical newspapers, bear in mind that the birth notices may not list the name of the child. They might only say a son or daughter.
Look for what papers may have been published in the area by using the US Newspaper Directory at The Library of Congress' Chronicling America website. Note the frequency of the paper, and take that into account when doing your searches.
It can help to choose a newspaper from the time period and read an issue from cover to cover, to get an idea of the layout of the paper, what kind of news is covered, the geographical coverage of the paper, the time lag between events and when they are written about in the newspaper, the style of how names are listed and the abbreviations that are used. Use the information you discover to help with your searches.
You may not get that 'lucky dip' in the newspapers or church records that was created near the time of the birth, but you may be able to get more information than you have now if you search widely. I have found information in the newspaper much later than the event itself via features that reprint information from long ago. It's common to see '25 years ago', '50 years ago' features. The longest gap I've seen so far in my own research gave news about a visit in the '150 years ago' feature.
Don't just look for birth notices. Look for any information about the family that might give you a clue about their church affiliation, which could help you identify the correct family when searching church records. For visits, look for items in the social news for both locations (for your case, both in Buffalo and in Pennsylvania) and in any locality where relatives lived.