Hi alicef, here are my findings...
- Petroselinum crispum with ruffled leaves aka "curly parsley"
- Petroselinum crispum var. neapolitanum with flat leaves aka "flat-leaved parsley" / "Italian parsley".
There is no such thing as "French flat leaf parsley". What is sometimes called "French parsley" is in fact a different species, Anthriscus cerefolium aka "common chervil". But chervil also exists in a 'curled' variety.
As usual with plants, the only valid reference is the scientific (Latin) name. The same common name can apply to 2 different species and the same species can have quite a few different common names. The extra advantage of using the scientific name is that it's international. So using scientific names for plants on a gardening site does not mean being pedantic (as I've sometimes been reproached with), on the contrary, it makes things more clear. Of course, asking for a bunch of Petroselinum crispum at the greengrocer's market stall, that would be pedantic ... and ridiculous.