It's up to you whether you buy it before or after you join - it's a bit pointless lugging a £2 bucket across the country by train from Yorkshire if you don't have to, but if you'd rather, then it's your choice. The AFCO list on your pre-joining letter is compiled by Lympstone & advises what to take to Lympstone. The recruit handbook is for guidance & information.
All will become clear when you have your pre-entry brief at your AFCO if you haven't already.
When I joined we were all issued a red bucket with white lid. I cannot confirm but i would presume this is still the case. It would look very messy if we all had buckets of different sizes and colours.
Also the issued bucket was not very big. The idea being you do your dhobi daily so do not need a large bucket.