The way I've always interpreted the rule is as follows: the only way to prevent the burden is to wound the ring bearer. In the end you didn't wound the ring bearer, so you didn't pay the cost required to prevent the addition of the burden. Therefore the burden gets added. As mentioned,
Strength of Spirit is worded perfectly: you did still exert (it should trigger Demoralised etc) but the act of exerting did not require you to place the token.
The one time where it gets a little hazy is where the word 'instead' is used instead of 'prevent' (mainly the One Ring adding burdens 'instead' of wounds). I believe that Decipher intended
Faramir, Bearer of Quality to be able to use the ring to pay the cost of his action (as it would surely have required an exertion rather than a wound otherwise). But I think Gemp rightly does not allow that: the wound was not placed, so the cost wasn't met. If that did count as a wound,
Promise Keeping would be so much better than it already is!