Unfortunately, since no errata or FAQ is out I need an official ruling on this. I made a call on it myself, but two players took it assuming otherwise and three players heavily disagree with my call on it, so I fear I need an official decision.
Quick Healer
Cost: 10 BP
A Quick Healer recovers from damage more quickly than other characters. The character receives a +2 dice pool modifier to all Healing Tests made on/for/by him, including magical healing.
Here, the "by him" part can be taken two ways. The first is tests taken by him for his own benefit, so basically Resting and any touching yourself with FA/Medicine/Heal. The second is that it applies when you use FA/Medicine/Heal on others. However, the fluff line makes clear that this isn't about treating others, it's about your own recovery abilities, which work both when people treat you and when you simply rest. This interpretation also fits with the name. It's not Excellent Healer or anything, it's Quick Healer, you heal quickly.
Quick Healer
Cost: 3 Karma
A character with the Quick Healer quality receives a +2 dice pool modifier to all Healing Tests made on/for/by her, including magical healing.
Here, the fluff line has been dropped. So has the cost, significantly. Since the fluff line isn't there, some people assume it includes tests made by the character on others. Given how it's been made cheaper, while other ways of getting a +2 are both more expensive and more limited*, I do not believe the SR4 intent has changed. However, without the SR4 fluff, all I have as support are the extremely cheap cost**, the name and a comparable situation, namely Symbiotes***. Unfortunately this is not enough to convince my Missions players, causing bad blood because I made very clear that I am not giving them a +2 when they Heal or First Aid others.
*: Specializations and Mentor Spirits both only give a +2 on either the First Aid or the Magical side of the picture, and are 5~7 karma. In comparison, 3 karma is real easy.
**: I paid 6 karma after chargen for it myself, since as Street Sam you can never have enough dice to recover from damage.
**: Symbiotes note "as a dice pool modifier on healing tests", not stating it's only your own either. However, given how they're in your blood, helping you, it's quite clear the intent is that it only works on yourself. So a lack of full clarity is not something limited to just Quick Healer, supporting that it's quite likely the Quality was simply not written to resist confusion.