refactor(browser): migrate logs to shared extension lifecycle (#4533)

* refactor(browser): migrate logs to extension lifecycle

* docs(browser): note remaining logs disposal cleanup

* refactor(browser): pass shared client to lazy logs

* fix(browser): support historical logs capture names

* docs(browser): describe logs compatibility fix

* docs: split browser logs changesets

* refactor(browser): remove legacy logs initializer

* test(browser): cover logs in slim bundles

* fix(browser): address logs lifecycle review comments