Hey Aiea,
The vSphere Client, VI Client etc all point to the C# client included in the default vSphere/vCenter install bundle.
On the other hand, the Web Client is different.
http://blogs.vmware.com/vsphere/2012/11/which-vsphere-client-should-i-use-and-when.html
The below link shows the files involved
https://my.vmware.com/web/vmware/details?downloadGroup=VCL-VSP510-VC-510B-EN&productId=285&rPId=3362
The vSphere Client or the VI client is the one below
VMware-viclient-all-5.1.0-941893.exe
File size:348M
File type: exe
Release Date:2012-12-20
Build Number:941893
VMware vSphere Client 5.1.0b
VMware vSphere Client installer. Note: if upgrading from a legacy VMware vSphere Client, the filename will have _5.1 appended to it.
MD5SUM:641781613bea5b6d229c5510d82788de SHA1SUM:e1e5e9d2ef51e105659cd881197a287f318ef3e3
Let me know if this helps
Regards
a