Update the example README to the payload the target sends today

[PetrHeinz]

The JSON samples in example-project/README.md date from the Logtail days: typed field names (logger_string, level_string, line_integer, message_string …), numbers as strings, and a message rendered by the old default layout (2022-01-26 10:25:06.0980|DEBUG|Program|…), none of which the target sends. Docs only, no tests.

• Every sample is replaced by the payload 1.1.0 sends for the same call: dt, message, level, and context with logger, properties and runtime. I captured them by running the example Program.cs (the color formatter left out, as the README's setup does not use it) and a ShoppingCart class with the README's nlog.config against a local receiver, on NLog 4.7.11, which the example project gets through the package. Only the file paths are replaced with the Windows path the old samples used.
• A new sample shows an error logged with an exception, with the top-level exception field.
• The "Filter logs" section no longer shows a layout-rendered message line: with layout="${message}" the logger name is not in the message, it is sent in context.logger. The search formula uses context.logger.
• Fixes the "Logtai!" typo in the logging example and "sent from to" below the search formula.

The changes stay in the JSON samples and the sentences around them. The "Additional configuration" snippet and the endpoint lines, which #19, #22 and #23 edit, are untouched, so they merge cleanly with this PR.

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