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The term to use is 'tenants in common in unequal shares' as this is often used along with a 'deed of trust' which highlights how everything is paid and what share everyone gets.
For example, if it is a 1/3 share each, then you get 1/3 share of sale proceeds. If however, the deposits are unequal, and some people are paying more towards the mortgage, you can specify a set initial amount, and the amount of funds everyone gets is determined by an algorithm, subject to contribution towards mortgage and running costs. Conveyancing solicitors don't like dealing with anything more complicated than a standard split, so have a discussion with your purchase solicitor. Technically buying in tenants in common should not cost any extra, and solicitors often have a flat fee to deal with a deed of trust along with a purchase.
That is definitely the best thing.
Just remember they usually charge by the hour. Having an agreement agreed in principle between you all in the key points will speed things up, and hopefully reduce cost. That said an independent 3rd party presenting options can feel less adversarial.
The Solicitor should have proformas for all this stuff, but it's always negotiations/amendements that take time and cost. The writing part tends to be easy. I'd make sure you all go prepared.