It is in this context important to recall what a data controller and a data processor is. The data controller is the person or entity responsible for the data processing. They determine the purposes and means of this processing. A data processor merely performs data processing on the controller's behalf.
The processor must not process data for their own purposes, but can make low-level decisions about the means of processing. For example, a controller may specify the purpose of ensuring the security and integrity of the services, or the purpose of creating basic statistics. The processor might implement this in part through the means of keeping a logfile. Often, such purposes are agreed upon via a data processing agreement or the terms of service, leading to the legal fiction that the processor was instructed by the controller.
In your scenario, the controller is presumably not subject to the GDPR1, and therefore does not need a clear purpose or a legal basis for their processing activities. However, the EU-based data processor is bound by the GDPR, and can only process data as instructed by the controller. The processor has to comply with the GDPR with respect to how they carry out the processing activities, but the processing activities themselves don't have to be GDPR-compliant. Thus, it might be perfectly fine for an EU-based hosting provider to keep logfiles with IP addresses, even if those logfiles wouldn't be GDPR-compliant.
1: Whether GDPR applies to a non-EU data controller depends not on whether processing occurs in the EU, but on whether the controller is offering goods or services to people in the EU, or observing behaviour of people in the EU – see Art 3(2) GDPR.
The cited guidelines continue to enumerate which provisions of the GDPR would still apply to EU processors working on behalf of non-EU data controllers (pages 12–13). These requirements are about how the controller and processor work together and relate to internal compliance of the data processor, but they do not affect the processing activities themselves.