"Regular" (voluntary) redshirt - you can't play at all in a season. Athletes are given 5 years to play 4.
"Medical" redshirt - common term for a medial hardship waiver. CyCy sounds correct, but I can't vouch for the details. If you did NOT take a regular redshirt season, which means you are STILL within the normal 5 years to play 4; and you were out most of a season (the percentage of the season is the detail I'm not sure about), then you qualify for a medical hardship waiver and are allowed to play your 5th year, even though you've played all or part of 4 seasons already.
6th year of eligibility - I'm much less certain about this, but I believe it's basically getting 2 medical hardship waivers -- if you miss 2 seasons to injury you might be given a 6th year to play your 4th full season.
Bundrage doesn't qualify for a medical hardship waiver (or a 6th year of eligibility for that matter) due to taking a regular voluntary redshirt season. Of course anyone can apply, but the argument will pretty much be "pretty please with sugar on top because it was only the 4th play of the season (if I remember)".