Hi Mathieu,
I have exactly the same issue that you are facing. I have a macbook pro late 2011-8,1 model. The Boot Rom and SMC version currently present is different from that recommended by Apple for late 2011 model. ( Current version is MBP81.0047.B2A for boot rom and 1.68f99 for SMC . Recommended version is MBP81.0047.2AB and 1.68f98 (SMC1.5) ).
2 years back my laptop started responding slow, a few months after my warranty got over!! After that, I did not use it for a long time. Two week back, I changed the hard disk to SSD and reinstalled all the OS from scratch ( EI Captain). Once this was done, my charger is not charging properly. When I attach it, green light come up for sometime and then suddenly goes off resulting in no charging. After trying different solutions from various sources, which included trying another mac charger ( When i tried this, battery was charged to 45 percent and then the new adapter started turning off. So I understood it is something to do with the firmware or some purposeful manipulation or battery issue or heat issue), resetting SMC, going to sleep mode and then starting again, battery remove and replace ( No battery issue is present),cleaning fan, checking heat and so on finally I observed that the bootrom and smc version is not the one meant for Macbook Pro late 2011- 8, 1 model, instead the one which was meant for Macbook pro Early 2011 -8,1 was installed. I suppose this installation came as part of the EI Captain as this was the only change I did. I could almost conclude that this is an issue with firmware update because, even though it shows not charging from adaptor, the battery keep remaining at 10 percentage without reducing. It only reduces if I disconnect the adapter.
I think this is some purposeful manipulation done by Apple for us to buy their new products or some mistake ( less chance).
Please let know if you have sorted out some way for this issue.
Thanks
Soorej