I'm going to 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 simply a representative of stakeholders who has authority to make decisions on the priority of work in the Product Backlog. The Product Owner isn't a technical position. It is a business position. The person in the Product Owner role needs to understand the customer and their needs. After all, their primary function is to write and prioritize the Product Backlog Items, to answer questions and coordinate with stakeholders, and to 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 for more info). Those roles are at odds with each other. The Product Owner is responsible for driving the business needs while the Scrum Master is responsible for the team. Sometimes, the needs of the business is 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?
There's nothing that prevents the Product Owner from attending the Daily Scrum meeting. The Daily Scrum is the opportunity for the Development Team to self-organize around the day's activities and for the Scrum Master to coordinate with the team on what issues need to be resolved to not slow development. 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 from outside the Development Team who attend the Daily Scrum do not disrupt the Development Team's process for the meeting.
The time immediately following the Daily Scrum can be used for more in-depth discussions among the Development Team members or to adjust the plan for the Sprint. People from outside the Development Team can participate in this activity.
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 would agree that the customer interaction aspects of the Project Manager role do map into the Product Owner role. However, other aspects of the Project Manager role, such as managing the schedule and budget, are more of a Scrum Master function. Other project management functions, like estimating, managing risks, and internal communication, belong to the entire Development 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.