Update example doc to support nextJS versions 13/14

[yansavitski]

Update doc with integrations search ui and nextJS 13/14 versions.

Acceptance criteria:

• [ ] Example work from the box with 13/14 version of next js
• [ ] Works with app directory in nextJS
• [ ] Work with page directory

[Samiul-TheSoccerFan]

Adding references as these two issues/threads might get direct benefitted from this change:

https://github.com/elastic/search-ui/issues/1004

https://discuss.elastic.co/t/in-responsive-mode-when-i-select-some-filter-the-filter-automatically-closes-withoutin-responsive-mode-when-i-select-a-filter-the-filter-closes-automatically-without-me-clicking-save-filter-clicking-on-save-filter/353148/9

[botelastic[bot]]

This issue has been automatically marked as stale because it has not had recent activity. It will be closed if no further activity occurs. Is this issue still important to you? If so, please leave a comment and let us know. As always, thank you for your contributions.