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:

#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]

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