So, I've actually done a bit of research into the etymology of a few of the spells in Harry Potter due to a fanfiction I'm writing. Several of the spells are Latin, yes, but there are also spells (such as Episkey, referenced above) that are Greek or other ancient languages in origin.
A few that I've read about included Colloportus, which is a compound word that could have been derived from either the Greek collo or the Latin colligo, and the the Latin portus. There's another example of this in the Levitation Charm, Wingardium Leviosa where we have wing an English word, arduus or arduum, Latin, with the -ium Latin ending, and levo and levis in Latin all forming parts of this composite word.
Then you have the Muffliato charm which is a derivative of English, "muffle". There are also much, much simpler spells. For example, pack.
I think it's worth mentioning that most of the Latin/Greek spells were invented around the 1600s. This makes sense as the Renaissance in Europe was between 1300 and 1700, and if the Muggles were being enlightened, of course the Wizarding population would be reaching new heights as well. It also bears mentioning that this is the time period when the Statute of Secrecy was implemented because of all the attention the Wizarding population was accruing. During this era Latin was commonly used in Europe by scholars and in the Church. Because of this, it was one of the languages used most probably because in order to form an opinion and contribute anything back into your field of study, you needed to use the language that group of people on the wider spectrum used; if they had used Middle or Old English, the spell would likely only be understood in England. Using Latin or Greek in that time period allows people in France, Germany, Italy, and in fact most of Europe to not only understand their work, but use it and ultimately give feedback on it. This same principle is applied even to this day; we use English in this forum because it is the most widely understood language in which to compare ideas.
Latin was the English of their era.