Use MPSKit's logging macros and adapt_solver

[leburgel]

MPSKit 0.14 allows LoggingExtras 1.2, which deprecates @infov, @warnv and withlevel(; verbosity). MPSKit 0.14 also deprecates updatetol in favour of adapt_solver. Under Pkg.test, these deprecations print 136 and 65 warnings respectively in the Julia 1 ubuntu jobs of run 38291840013.

We now log through MPSKit's with_verbosity and its @log_* macros, and drop the LoggingExtras dependency. CTMRG verbosity levels are unchanged. The BP init message and the simple-update header and timing lines move from verbosity 1 to 2.

🤖 Generated with Claude Code

[leburgel]

Also force merging this one, since it's approved and test failures are resolved on main.

[codecov]

Codecov Report

❌ Patch coverage is 83.33333% with 4 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/algorithms/optimization/peps_optimization.jl	0.00%	3 Missing ⚠️
src/algorithms/bp/beliefpropagation.jl	80.00%	1 Missing ⚠️

Files with missing lines	Coverage Δ
src/PEPSKit.jl	100.00% <ø> (ø)
src/algorithms/ctmrg/ctmrg.jl	80.68% <100.00%> (ø)
src/algorithms/time_evolution/simpleupdate.jl	96.35% <100.00%> (ø)
src/algorithms/bp/beliefpropagation.jl	93.93% <80.00%> (ø)
src/algorithms/optimization/peps_optimization.jl	35.35% <0.00%> (ø)

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