First things first, you need to figure out what's the tonic. Probably, it's C in your example. From there, you can think about modes in two ways, parallel and relative. Parallel modes are when C is still the tonic, but the scale around it changes. Relative modes are a useful concept when C remains tonic, but there is a strong emphasis on another pitch. Let's say C remains tonic, and you keep hovering over a D minor chord. Then, it might make sense to speak of it as the relative Dorian. You are still in C Major/Ionian overall though. You can tell by B wanting to move to C, by D wanting to move to C or E, F wanting to move to E (mostly) or G, and so forth. Why then does it make sense to speak of relative modes at all? It's because if you keep playing a D minor chord in C Ionian, you won't only be hearing those pitches against the tonic C (that we keep in the back of your minds), but also as extensions of the D minor chord. F won't be willing to move to E as much as if you were playing it over a C chord, since it's consonant with the underlying D minor chord. On a more global level though, the D minor chord itself wants to resolve to a C major one, in this particular case, and that's how you are still in C Major overall. If D minor itself is the tonic, and your notes are CDEFGAB, then it's absolutely D Dorian. If C is the tonic, it's the relative Dorian at best, but you are in C Ionian overall. Makes sense?

Most of the confusion about modes comes from mixing up relative and parallel modes. For example, there is the Phrygian Dominant mode, 5th mode of Harmonic Minor. That name makes a lot of sense if you see it as a relative mode of Harmonic Minor. If you have G Phrygian Dominant and G is the tonic though, there is nothing dominant whatsoever about this mode. In that context, it might make more sense to call it by another one of its names, like Ahava Rabbah or Freygish. You'll hear it referred to as Phrygian Dominant all the time though.

Whenever a piece is in a mode that's not classical Major (basically Ionian) or Minor (mostly a mixture of Natural, Melodic and Harmonic Minor), and it has a clear tonic, it falls under something called Extended Tonality. There is also such a thing though as playing modes without a clear tonic, such as in some pieces of Modal Jazz. You could think of that as pure relative modes without an overarching main mode

In your example, it's all straight C Ionian, no relative modes, nothing. If you arpeggiate a chord like F A C A F, then make some movements such as F G A C (G resolving to the third of the F major triad?) or F A C E F (E resolving to F instead of the other way round?), one could maybe call that the relative Lydian, and that's as far as it gets. You're still in C Ionian overall, unless you have somehow modulated