There is also something that I note... in the 1900 US Census, William's brother John b. Ohio in 1886, yet father migrated in 1888, so that's two discrepancies.
One thing that you might do by process of elimination.Try to follow William and Agustus's son William, that is, without regard to yours.You seem to have the US Census, so I also see a William in Hamilton County with both parents b. in Germany later.
I would also search "Smith" on the Lorain County page at http://www.USGenWeb.nethttp://www.USGenWeb.net (select Ohio and then the county) and see if you can find the cemetery listings.Many of their cemeteries are online.
You sound like you have access to the US Census, so my looking further for you is probably a duplication of effort.If you have an Ancestry.com subscription, you might also look for their immigration record, as Ancestry.com's immigration search capabilities are growing.
Regarding finding a birth, I believe there are two indexes available for sale by the Lorain Chapter of the Ohio Genealogical Society (member) that you can access at the same website (above).
I think persistence will be your key, and with the mass number of people with the similarity of names "William Smith," verification would especially in order.
I'm sorry I could not help you further.Best regards and luck.