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Hello Vladislavs Barkovskis,
Per the description shared, I understand you want to set up the List Web Part with the view displayed in your second screenshot, however, when I tried to test the behavior at my end with the custom view, I can get the similar view as your first screenshot, see below:
Moreover, checking the List web part settings and enabling the following options will give the following view:
In addition, as you are not having +New, Share, Export to Excel commands in your second screenshot, I would like to know whether you customized your SharePoint List via PowerApps or any other source? If you go the List, do you have the same view shown in your second screenshot? If yes, can you please share the view settings with us?
Appreciate your understanding!!
Best Regards,
Stacey