We had a I, vi, iii, V major key progression in the key of D major. It is now a I, vi, bVI, V progression. It is still in the key of D major. The borrowed chord is not a modulation, merely a temporary jump to an out-of-key chord.
Note, just as with the bVII in Example 1, this borrowed chord, within its home key of D minor is a VI chord. The 6th scale degree of the D minor scale is the note Bb and so the (major) chord built from that note must be a VI in Roman numerals. When that same chord sits within a D major progression as a borrowed chord, it must be labelled as a bVI. The 6th scale degree of D major is a B natural which we must compare and reference this borrowed chord against. Bb is a flattened 6th so Bb major is a bVI chord.
Here are the two mp3 audios again, side-by-side for easier comparison.
Also, here are the TABs for both progressions.
