The partnership must make the payment on or before the due date of the partnership tax return, not including extensions.
Does the 401k provider care or is in on the partnership accounting and tax professional to make sure that everything is documented correctly?
No, the trustee of the plan is not usually involved in the partnership accounting or tax preparation.
If a partner would like to make an allocation to a 401k for 2013, how do you document that a payment was made or is intended to be made?
There is not specific language or methodology required to be used and the partnership operating agreement may place requirements or restrictions.
in general, this decision will be documented like any other partnership decision (compensation, investments, acquisitions, expansions, etc.) Unlike a corporation there is not an annual meeting, so these financial decisions are usually documented in memos or agreements (if not already part of the operating agreement).
And, as usual, the check or electronic transfer of funds will be the documentation of the actual transfer of funds. Generally, an entry on the partnership books is made to express intent to make the payment (and record the liability that will be paid later).
Please understand that the election to defer has to be made before the end of the year, even though the actual payment is made by the return due date. For more information see http://www.irs.gov/pub/irs-tege/forum08_401k.pdf
Please ask if you need more discussion or clarification.