It's important to remember that Redington's conditions for immunisation are very much a "gold standard" for assetliability matching. The conditions are as follows:

PV(assets) = PV(liabilities)

Vol(assets) = Vol(liabilities)

Conv(assets) > Conv(liabilities)
Now, these conditions are heirarchical in the sense that if (1) isn't achieved, then it doesn't make much sense to check (2). Likewise, if (2) is not achieved, then it doesn't make much sense to check (3). If your portfolio has met (1) and (2) but not (3), then the matching is not bad, but it's not the gold standard.

Returning to the question: "...sometimes the solutions equate the DMT, sometimes they equate Vol, and sometimes they just equate the second derivatives of the PV."
So this part is talking about how to check condition (2). I think that you meant to say "just equate the first derivatives of the PV" because second derivatives have to do with condition (3).
Note that when the current interest rate \(i\) is known, there is a very simple mathematical relationship between DMT and Vol, namely: \(\tau = (1+i)\nu(i)\) in the notation of Chapter 14, and this relationship holds for both assets and liabilities. So it doesn't really matter: you can equate DMTs or Volatilities.
If you have already satisfied condition (1), as you should have, then \(PV(assets) = PV(liabilities)\) and you can get away with just equating the first derivatives, because \(\nu_{assets}(i) = \frac{PV(assets)'}{PV(assets)}\) and \(\nu_{liabilities}(i) = \frac{PV(liabilities)'}{PV(liabilities)}\).

Lastly, you should always show that condition (3) is met. Maybe in that situation, it was just easy to see by general reasoning that the convexity of the assets was greater (in which case you should write down the reasoning!)