Domoticz

Question:
How to easily control lighting:
Lamp - switch in domoticz - Sonoff 4CH
Pir detector - Detector in Domoticz - integration with Satel Integra
Local momentary switch on the wall connected to Sonoff 4CH connected to a button on Sonoff.

How to implement control so that the detector turns on the lamp for a specific time and the switch turns on permanently.

How do you do it at home? By blockly or Lua or dzVents

Check the wiki, search this forum. Search for light motion and you will finds lots of examples.

I prefer communication between devices.
PIR detects movement and sends a command directly to light switch to temporarily turn on the light.
The light switch knows that it has received a command from the PIR sensor and therefore turns on the light for a certain time. It also sends the information that the light is on to the home automation system Domoticz
After the set time has passed, the light switch turns off itself and sends a message about its status to the Domoticz again.
This will work even if your PI with Domoticz installation is removed to silicon heaven.
But you need to choose a device that allows uploading custom firmware or something like Shelly Plus with scripting capability.
Or run everything through Domoticz...

I coordinate Z2M PIRs, push buttons and Milight bulbs with a dzVent script.

Light is turned on by PIR if illumination is lower than a given value (to avoid turning it on during day), and turn it off after a delay.

Pushing button inverts bulb state, and lock PIR for 30 seconds (to avoid turning bulb on by PIR just after turning it off by push button, as I have large rooms with 2 or 3 PIR).