Use ESP-01 with Arduino Uno/Nano, with the ESP8266WiFi library There is this library - https://github.com/esp8266/Arduino
That provides a more user-friendly API for WiFi capabilities with the ESP8266.

However, all tutorials that I found so far for this library show how to program the ESP8266 (usually 12) directly. That is not what I'm looking for.

My current setup is having an Arduino Uno/Nano where the ESP8266 is connected to, say, GPIO8 and GPIO9 (RX, TX) and I use SoftwareSerial to communicate with it with AT commands. So I want to keep all the logic in my Arduino, but use ESP-01 only to provide WiFi capabilities to my Arduino.

However in that library I don't see the option to set the serial port pins (RX and TX) so I'm not sure if that's even possible.

In short - I want to use this library, but not program the ESP directly. Also, I'm using an ESP-01 and not 12.

Thanks.
You cannot use that "library". Simply because that isn't a library, that is a core. It's the entirety of the Arduino API designed to run on an ESP8266.

You can use it to program the ESP8266 directly and nothing else. No part of that has anything at all to do with Arduino boards like the Uno.