Is it possible to have the start of the rental to be only within 4 days from the moment of booking, but without restrictions for the end of the rental?

Yes. This can be configured.
1. Copy the rule of work and open its settings.
2. Unset the Show in the legend checkbox.
3. Set the checkbox Shift end from the current moment, specify 4 days and save the changes.
4. Add a reaction.
5. Select the type Error and specify a message about the 4 days booking restriction.
6. Set the option Enabled in all rules.
Enable this reaction only at the old rule of work.
As a result, when customer books the time at the intersection of the rules, the reaction will not run. And when he is booking only on the old rule (e.g. from the 5th to the 10th day from the current moment), an error will be displayed.
See an example in action:
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