I will answer the questions in a slightly different order than you asked them since I think it helps to build up the answer.
Product Owner is the person who is customer. A team member can be nominated as a proxy of product owner?
A Product Owner does not have to be a customer but a representative of stakeholders who has the authority to decide on the order of work in the Product Backlog. The person in the Product Owner role must understand the customers and their needs. Their primary function is to create and prioritize the Product Backlog Items, answer questions and coordinate with stakeholders, and review the incremental deliveries to ensure that they meet the needs of the users in the intended environment.
One piece of guidance is that the Scrum Master and Product Owner should not be the same person (see here and here). These roles are at odds with each other. The Product Owner drives the business needs, while the Scrum Master is responsible for the team. Sometimes, the needs of the business are at odds with the needs of the team.
What is role of Product Owner in daily Scrum, if he/she is part of this meeting?
Nothing prevents the Product Owner from attending the Daily Scrum meeting. The Daily Scrum is the opportunity for the Developers to self-organize around the day's activities and for the Scrum Master to coordinate with the team on what issues must be resolved so as not to slow development. Unless there are specific questions, there's not much for the Product Owner to do at the meeting, though, other than observe the team. The Scrum Master ensures that people who attend the Daily Scrum do not disrupt the Developers' process for the meeting.
The time immediately following the Daily Scrum can be used for more in-depth discussions among the Developers or with other stakeholders or to adjust the Sprint plan.
Can we say that a Project Manager who directly interacts with customer is nominated as proxy of Product Owner?
There's no direct mapping between a traditional Project Manager and the roles in Scrum.
I agree that the project manager role's customer interaction aspects map into the Product Owner role. However, other aspects of the Project Manager role, such as managing the schedule, are more of a Scrum Master function. Other project management functions, like estimating, managing risks, and internal communication, belong to the entire team.
When we use term "External Stakeholders Engagement", are we referring to Product Owner or something else?
Most likely the Product Owner, yes.
The Product Owner is the person who speaks on behalf of the customers and users of the software being developed. The Product Owner needs to understand their needs to be able to write user stories, answer the team's questions about those user stories should any arise, prioritize the Product Backlog, and more.
The Product Owner isn't expected to know everything, but they do manage communication with the outside world. If they don't have an answer, they know who to engage to get the answer to the development team.