docs(proxy): explain how team_member_budget updates reach existing members

[devin-ai-integration]

Summary

team_member_budget was documented as "set a max budget for a team member" with nothing about who a later change applies to. This spells out the model that BerriAI/litellm#41347 restores: the value is one team-wide default that every member added without max_budget_in_team follows, so /team/update (or the Default Budget field in the team UI) reaches those members, while /team/member_add with max_budget_in_team or a later /team/member_update gives a member their own budget that stops following the default

Touches the Team Member Budgets section in docs/proxy/self_serve.md and the "Update a team member's budget" section in docs/proxy/users.md, cross-linking the two

Companion to BerriAI/litellm#41347, which fixes the regression where /team/member_add copied the default per member so /team/update silently stopped applying to existing members (LIT-7716)

Link to Devin session: https://app.devin.ai/sessions/5d1eee9e522749968ae46d78e845be94
Open in Devin Desktop: https://app.devin.ai/desktop/session/5d1eee9e522749968ae46d78e845be94?variant=devin

[devin-ai-integration]

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Project	Deployment	Actions	Updated
litellm	Ready	Preview	Sep 16, 2026 1:27am UTC