Someone has confused you.. probably a banjo player. Just kidding.
F can be played two ways: first is the barre shape using your first finger to cover the first fret, then the "E" shape.
OR, you can fret the 6th string with your thumb, then your first finger will cover the 1st and 2nd string, you middle finger takes the 3rd string, your pinky takes the 4th string and your ringer finger the 5th.
Banjo players use a "open" F which I believe is Ralph Stanley inspired. I like the sound of it. Like this: 03210. You can always experiment and tune your guitar to Open G: DGDGBD.
An "open" F on the guitar in standard tuning is actually Fmaj7, the open E being the major 7th of F, a completely different sound.
A C to Fmaj7 (I to IVmaj7) change is something that I've heard in a few rock ballads from the 60s and 70s. Since both chords share the E note, it's acting as what is called pedal tone... a note that sustains over an entire progression. This gets back to voice leading.. the E is bringing your ear back to the I chord, either from the IV or V.