What is the ideal size SSD to use in replacing the existing HDD on my MacBook Pro?

The size of the SSD doesn't really relate to what the machine is capable of doing. It relates to how much data you have to store. Usually a big factor is how many photos and especially videos you have as those are things that take up a lot of space.

Note the drive prices in https://eshop.macsales.com/shop/ssd/owc/macbook-pro/2010

The 1TB drive isn't much more than the smaller ones and islikely larger than what you had before

For what you may have had before, see the following from MacBook Pro (15-inch, Mid 2010) - Technical Specifications

320GB or 500GB 5400-rpm Serial ATA hard drive; optional 500GB 7200-rpm hard drive, or 128GB, 256GB, or 512GB solid-state drive

That tells you to size of the drive but not how much of it you actually used/needed. But based on that, if you weren't having issues with lack of space and your future needs are the same as in the past, 500GB would be adequate. But if in doubt, you can double that to 1TB for less than $20 more.

That depends on how much storage space you need. What would be ideal for me would not likely be ideal for you.

One place to start would be to look at how much storage space you are currently using.

Also, the following is a well-regarded source of SSDs: https://eshop.macsales.com/shop/ssd/owc/macbook-pro/2010

if you surf to:

https://eshop.macsales.com/

and use the link in the header labeled "my Upgrades" and

enter your computer information,

the site will show you everything that works, first time every time, with you exact model Mac.

look carefully at the prices as the size increase. These drives have been available for a while, so there is likely some "price-compression" up to a certain level. For example. a 1,000 GB may be only a few dollars more than a 500 GB.

default drive when new was :

"a 320 GB to 500 GB Serial ATA (5400 RPM) hard drive,"

--from EveryMac.com

[in my opinion] you should not pay money for a drive smaller than 500-ish GB

500-ish GB to around 1,000 GB is a good utilitarian size, re-usable in another Mac or enclosure if you decide to re-purpose it later.

It depends on what you plan to do. For example I use graphic design software and hard drive fill quickly plus adobe recommend using external ssd for ram preview and storing assets like video clips also for premiere pro. This is supposed to take the stress off the internal ssd. I installed 1 TB ssd and also used some fast external like Samsung 980 pro 2 TB ( in an an Acadia 40 GB case). Works great all round. Just remember nowadays everything data is larger ( uncompressed RAW photos , UHD and HD videos, etc. 10 12 16 24 bit. All adds to file size. Hope this helps

Memory is not storage and the amount of memory you have has nothing to do with hard disk or SSD storage size.

Memory is used on a short term basis while the computer is doing computations and other operations. Storage is where your files, data, photos etc are stored on a long term basis.

Click Apple Menu > About This Mac nd look for the Storage size and how much it is in total and how much is available. Also see: https://www.lifewire.com/check-storage-on-mac-5092347

put that disc in the drive, power up and hold the D for diagnostics key until it loads the diagnostics.

Just what the instructions in the upper right say.

run the long version of the test is you have a choice. Once the diagnostic starts, do not despair if the screen seems to freeze, It is using every cycle to pound on the memory during the memory test, leaving none to update the display. typically takes under half an hour..

That is not an area that I have enough personal experience to offer advice. Contact customer support: https://eshop.macsales.com/Service/

They are very good.

.

Apple also lists it as a kk2010 model. See the following from Identify your MacBook Pro model - Apple Support

MacBook Pro (15-inch, Mid 2010)

Model Identifier: MacBookPro6,2

Part Numbers: MC373xx/A, MC372xx/A, MC371xx/A

Newest compatible operating system: macOS High Sierra

Tech Specs: MacBook Pro (15-inch, Mid 2010)

Don't know why the drive would be stamped 2013 unless it was replaced.

Did you install an OWC drive?

If so, they have customer support: https://eshop.macsales.com/Service/

and

install videos -- see: https://eshop.macsales.com/shop/ssd/owc/macbook-pro/2010

and

https://eshop.macsales.com/installvideos/macbookpro_15_unibody_mid10_hd/

and

https://eshop.macsales.com/articles/how-to-transfer-your-data-from-your-old-drive-to-a-new-drive

and possibly https://eshop.macsales.com/tech_center/formatting/Format_Mercury25_SSDs_Mac.pdf