Yes, the first progression would be in F#m when capo'd at the second fret and the second progression would be in E when capo'd at the second fret.
They are in different keys because the key is derived form the chords that are in the progression and not from the position of the capo.
In the first example Em is the key because Em is the strongest chord in the progression, it feels like 'home' when you play it. Adding a capo will take it up two semitones Em->Fm->F#m. Therefore the key is now F#m.
The same applies in the second progression. It starts in D and ends in D, coming back to D gives it a feeling of resolution, so the key is clearly D. Capo on the second fret will move it up two semitones D->D#->E. Therefore the key is now E.
Hope this helps Trev, but if not just say so and I'm sure someone more clued up than me will help you out.
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