Document that the termination modes receive the residual, not the increment

[ChrisRackauckas-Claude]

Please ignore until reviewed by @ChrisRackauckas.

Follow-up to #1149, which was closed after #1176 merged — but this half did not make it in. It was pushed to the same branch shortly after the merge, so master still carries the incorrect claim in two places.

The mode docstrings say \Delta u is "the increment computed by the nonlinear solver". It is not: every solver in this package passes cache.fu, the residual. The naming is inherited from the step-based use of the same machinery in DifferentialEquations.jl.

The distinction is what #1149 was reported over. ‖Δu‖ ≤ reltol·‖Δu + u‖ is the textbook relative-step test when Δu is the increment, because Δu + u is then the next iterate — the formula was never wrong, it is being handed the wrong vector. Fed the residual it compares ‖f(u)‖ against ‖f(u) + u‖:

case	‖F‖	‖u‖	objective
residual small, u large	1e-6	1e+3	1.0e-9 — fires, but only because ‖u‖ is large
residual small, u tiny	1e-6	1e-9	0.999 — converged, will not fire
u ≈ -F	1e0	1e0	6.0e+23 — denominator cancels

So this documents the modes accurately and adds a warning pointing at #1149, rather than changing the semantics — that is a behaviour change for every user of those modes and wants a decision. The issue reporter has endorsed redefining them as a relative-residual test (‖F(u)‖ ≤ reltol·‖F(u₀)‖) and additionally suggests a gradient-based criterion (‖Jᵀ F‖) for least-squares, which is the Nocedal & Wright convention and does not degenerate when the residual is nonzero at the solution. Neither is in this PR.

Also documents that reltol has no effect on a NonlinearProblem, whose default mode is absolute — verified: reltol ∈ {1e-1, 1e-3, 1e-8} gives identical iterations (5), f-evaluations (5) and residual (4.75e-14); only abstol moves them.

Docs only, no behaviour change. Runic clean.

🤖 Generated with Claude Code

Follow-up: SciMLStyle contract documentation

Commit 35e11e99 expands the generated termination-mode docstrings without changing solver code or termination behavior. The docs now use r = f(u) in the criteria, document the cache arguments and Boolean result separately from solver return codes, list internalnorm and every safe-mode keyword/default, describe Success/Unstable/Stalled/StalledSuccess, and provide examples for all nine modes. It also corrects the norm constructor signature and the termination-guide init example.

Focused verification:

• all nine modes constructed and solved a nonlinear problem with ReturnCode.Success;
• generated-doc inspection found # Returns and # Examples on all nine modes, # Arguments on norm constructors, and # Keywords on safe modes;
• Runic.format_string check: pass;
• typos lib/NonlinearSolveBase/src/public.jl docs/src/basics/termination_condition.md: pass;
• git diff --check: pass;
• isolated NonlinearSolveBase QA: 20/20 passed;
• focused Documenter build of the termination page: doctests and rendering passed. The repository-wide docs build was not used as a gate because it remained in the unrelated full-site doctest/render pass.