The chronologically third and fourth appearances I've found of the formulaic phrase 'close only counts in [something]' (and variants) do not mention horseshoes, but rather quoits. So, after "close does not count only in horseshoes" from the Lincoln Daily News (Lincoln, Nebraska)
15 Aug 1914 (paywalled link), comes this second appearance of the phrase concept in 1916:
Yes, there is a game in which 'close' counts. It is horseshoes.
Bisbee Daily Review (Bisbee, Arizona), 16 Apr 1916 (paywalled).
The third appearance I found is in The Union Postal Clerk of March, 1917, and this time quoits feature instead of horseshoes:
Day Rosemer — getting close counts in quoits only.
Then comes this fourth with reference to quoits from Evening Report (Lebanon, Pennsylvania), 10 Jan 1921 (paywalled):
It was close, but close only counts in quoits, and baby, how big these two points and the last remaining minutes of the fray looked to the Red and Blue.
That fourth appearance is followed closely by the 3 Oct 1921 appearance of the phrase with reference to horseshoes, as compiled by Doyle, Mieder and Shapiro in The Dictionary of Modern Proverbs (2012).
The association of the phrase "close only counts" (variants) with quoits was suggestive, given the much longer history of quoits than horseshoes, but searches of the UK and US popular press, and Google Books, did not disgorge any attestations featuring quoits earlier than the 1914 and 1916 appearances featuring horseshoes.