Your hard drive is too full...
doubling the RAM to 8GB needs more space. Your clue would be that 4GB leaves 4GB free space, then upgrading to 8GB needs THAT 4GB. So, the system memory space takes all of that. BUT, now there is 9GB reported as free, then your system is not using the memory correctly, "releasing" the 8GB that it needs for memory swap, therefore... you really have only 1GB free space, hence the "startup disk full" message. (Technically not really correct, but I think my explanation is relevant)
Your hard drive is too full... You need to get some space back. A good goal will be
10% of total space, plus the 8GB of RAM "space. So, I suggest that you should try for 40GB free. (Your 7-year-old hard drive is likely not helping you, being a mechanical device that runs almost constantly. Have you considered swapping out the hard drive for an SSD? That will have a much better result than
anything that you do with RAM, even if you upgrade your RAM to the full 16GB your 2011 MBPro can use. Just thought I would mention that.

)
If you REALLY would like to compare the 2 RAM upgrades, watch your Activity Monitor while using your Mac. The memory tab will show you how your processes and apps use the memory, and the memory pressure graph will show you how your RAM is impacting the system at the time. Green is good. Yellow is a lot of memory use, and Red will indicate that there is a lot of page-out activity, leading to a lot of slow downs, which will be much worse if the hard drive is low on space.