chore(tests): Update Playwright to 1.40.1 in browser integration tests

[onurtemizkan]

This PR updates Playwright version to 1.40.1 for browser integration tests.

• Removed deprecated usages like page.type
• Converted page.click event triggers to page.locator(...).click() aligned with the current docs.
• Refactored flaky textInput test (it does not seem to flake anymore)

There still are a few flaky tests I don't think they're related to this version update.

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser (incl. Tracing, Replay, Feedback) - Webpack (gzipped)	77.84 KB (-0.01% 🔽)
@sentry/browser (incl. Tracing, Replay) - Webpack (gzipped)	69.04 KB (+0.03% 🔺)
@sentry/browser (incl. Tracing, Replay with Canvas) - Webpack (gzipped)	72.93 KB (+0.01% 🔺)
@sentry/browser (incl. Tracing, Replay) - Webpack with treeshaking flags (gzipped)	62.68 KB (+0.05% 🔺)
@sentry/browser (incl. Tracing) - Webpack (gzipped)	33.06 KB (+0.08% 🔺)
@sentry/browser (incl. Feedback) - Webpack (gzipped)	31.25 KB (0%)
@sentry/browser (incl. sendFeedback) - Webpack (gzipped)	31.26 KB (0%)
@sentry/browser - Webpack (gzipped)	22.5 KB (0%)
@sentry/browser (incl. Tracing, Replay, Feedback) - ES6 CDN Bundle (gzipped)	75.57 KB (-0.05% 🔽)
@sentry/browser (incl. Tracing, Replay) - ES6 CDN Bundle (gzipped)	67.13 KB (-0.06% 🔽)
@sentry/browser (incl. Tracing) - ES6 CDN Bundle (gzipped)	32.97 KB (-0.08% 🔽)
@sentry/browser - ES6 CDN Bundle (gzipped)	24.43 KB (-0.06% 🔽)
@sentry/browser (incl. Tracing, Replay) - ES6 CDN Bundle (minified & uncompressed)	211.42 KB (-0.1% 🔽)
@sentry/browser (incl. Tracing) - ES6 CDN Bundle (minified & uncompressed)	99.44 KB (-0.17% 🔽)
@sentry/browser - ES6 CDN Bundle (minified & uncompressed)	73.08 KB (-0.08% 🔽)
@sentry/browser (incl. Tracing) - ES5 CDN Bundle (gzipped)	36.07 KB (-0.09% 🔽)
@sentry/react (incl. Tracing, Replay) - Webpack (gzipped)	69.45 KB (-0.01% 🔽)
@sentry/react - Webpack (gzipped)	22.55 KB (0%)
@sentry/nextjs Client (incl. Tracing, Replay) - Webpack (gzipped)	86.11 KB (+0.02% 🔺)
@sentry/nextjs Client - Webpack (gzipped)	50.4 KB (+0.03% 🔺)
@sentry-internal/feedback - Webpack (gzipped)	17.21 KB (0%)