* There was a big ancient civilization revival right about then. People were digging up bits of antiquity from Greek and Roman sites and hauling them home. France was fashionable then as well, but in a tarnished, scandalous sort of way. It was not as popular in the sort of upper crusty circles likely to establish an arts society. England fought with France a bunch... I don't recall any English/Ancient Greek battles...
Your statement that "The first use of 'philharmonic' was in London in 1813" is, um, somewhat less than correct. That might have been the first use in an English context, but in Ljubljana, Slovenia (which was then known as Laibach and was in the Austrian Empire) the Academia Philharmonicorum was founded in 1701. That translates easily as "academy of music-lovers" (or at any rate "harmony-lovers"), but the original members were all professional musicians, though their by-laws were amended to admit amateurs who were capable of making a contribution, and they gave regular performances. According to the history on the website of the Slovenian Philharmonic (http://www.filharmonija.si/), the organization was based on the example of similar societies in Italy, so it would probably make sense to look there for the introduction of the term (probably filarmonico or something similar), into modern usage, whether it was invented or borrowed at that time.
The Academia Philharmonicorum itself petered out during the 18th century, although the active musical life of Ljubljana certainly didn't. A new Slovenian Philharmonic Society was founded in 1795, and the orchestra called the Slovenian Philharmonic ('filharmonija' in Slovene, which might come over into English as "Philharmony") has existed officially since 1908. I think most modern "Philharmonics" are abbreviated forms of longer names (Philharmonic Orchestra, Philharmonic Society, etc.) in which 'philharmonic' still is really an adjective.
Yours in the service of facts that no one much cares about, Charlie Bowen