What is fastest operating system for my Macbook Pro 13" mid 2010?
You have to weigh the balance between increased functionality/usability/security of the new OS with the performance of the old.

You have to ask yourself the question "is the performance increase/decrease worth the added functionality?"

Better yet, "Am I going to notice the difference?"

Your CPU and GPU are not going to change so running comparative benchmarks against them with varying OS's tells you little.  If squeezing out every nanosecond of response is key to what you are doing, then this is entirely a moot thread; you should buy a new machine.

It seems that you are running into compatibility issues because your OS is now too old.  Upgrading to one that is pushing 4 years old now may not be the wisest move - especially if what you are doing centers around the web where the tech changes almost on a daily basis.

Here's my suggestion (and I have personally done this for a friend and they couldn't be happier) based on the fact you have a pretty good machine in your possession.

First, upgrade your RAM.  I'm suspecting you are seeing some performance slowdown if you are running out of memory.  Open up Activity Monitor and leave it open as you go about your day.  If memory utilization is close to or at the top of its limit, chances are you are sending the overflow memory requests to the hard disk (swap).  That'll slow you down for sure.

Officially, your Mac can to go to 8GB of RAM.  Unofficially, it can go to 16GB.  Get as much as you can afford.  More RAM means more space for your OS and applications to work in and less time being "swapped" to the hard disk.

Next, upgrade your drive to an SSD.  I can't begin to tell you how much a difference this makes not to mention battery life and less heat.  You are going to want to do a nice clean install, not an upgrade here.  

Finally (and this one is optional), don't throw away that old hard drive.  There are kits available that can turn your DVD drive bay into a second hard drive adapter.  You can have two HDDs in your MBP.  You can even opt for two SSD's and mirror them (RAID 1) for redundancy or stripe them (RAID 0) for increased performance.

One more thing....

If the "eye candy" and the animations are not important to you, those can be turned off, thus freeing up system resources to be used for other things.

Bottom line is this...for a relatively tiny amount of money (and some surfing on ebay or amazon) you can extend the life of your computer a few more years with much more recent OS.