I don't think it will show 'which' lens was used - only the focal length used.
I've just checked on a few of mine, and it appears to depend on the lens (and possibly camera) used.
So one taken on my A900 with the Tamron 150-600 shows 'Sony DSLR-A900' then '150-600 F5.0-6.3 SSM'
while one from my A6000 shows 'Sony ICLE-A6000' then 'E 55-210mm F4.5-6.3 OSS'
'Lens Model' appears to be an entry in the EXIF, and matches the description given at the top, so I woudl guess it is using that.