Instead of discussing Mann's planet, I'll draw an analogue with something simpler, but that works in the same way: a clock at the station, equipped with a calendar as well. Notice that both the astronaut at the station and the crew will need to agree on what reads on the calendar, even though 23 years go by for the astronaut while only a day goes by for the crew. Notice the issue is the same as with the position of Mann's planet, I'm just simplifying the situation so that we only need to deal with stuff at two places.

The key concept in here is that time is relative in a quite literal way. Time passes in the spaceship at a rate much different from the rate it passes at Miller's planet. To get things clearer, suppose the crew took a telescope with them down to Miller's planet and then looked at the spaceship so they could see the clock. What would they see? They'd see the days going by incredibly fast, because the spaceship time passes much faster when seen by them. Similarly, they see Mann's planet orbiting the black hole at a much faster rate than the spaceship sees. As a side note, note also that the telescope would only show the calendar a bit to the past, since it takes time for the light to reach the telescope. Also, there is no "good" definition of "present" in Relativity.

In summary, yes, once the crew returned to the spaceship, they would all agree on the position of the planet, but disagree on how long it took for it to complete each lap, similarly to how they would all agree that 23 years have passed on the calendar, but disagree on how long it took for the days marked on it to go by.

Instead of discussing Mann's planet, I'll draw an analogue with something simpler, but that works in the same way: a clock at the station, equipped with a calendar as well. Notice that both the astronaut at the station and the crew will need to agree on what reads on the calendar, even though 23 years go by for the astronaut while only a day goes by for the crew. Notice the issue is the same as with the position of Mann's planet, I'm just simplifying the situation so that we only need to deal with stuff at two places.

The key concept in here is that time is relative in a quite literal way. Time passes in the spaceship at a rate much different from the rate it passes at Miller's planet. To get things clearer, suppose the crew took a telescope with them down to Miller's planet and then looked at the spaceship so they could see the clock. What would they see? They'd see the days going by incredibly fast, because the spaceship time passes much faster when seen by them. Similarly, they see Mann's planet orbiting the black hole at a much faster rate than the spaceship sees. As a side note, note also that the telescope would only show the calendar a bit to the past, since it takes time for the light to reach the telescope. Also, there is no "good" definition of "present" in Relativity.

In summary, yes, once the crew returned to the spaceship, they would all agree on where they see the planet (I phrase it this way because, as mentioned, there is no suitable notion of present in General Relativity, so it doesn't make a lot of sense to say where the planet is, but it makes sense to say where they see the planet), but disagree on how long it took for it to complete each lap, similarly to how they would all agree that 23 years have passed on the calendar, but disagree on how long it took for the days marked on it to go by.

As for the parallels with Earth's view, similar effects will happen with astronauts on the International Space Station, for example. When they come back to Earth, we'll all agree on where we see Mars, but we might disagree on how long each lap it took around the sun took. Similarly, we'll all agree the rover Perseverance is on Mars, but we may disagree on how long its trip took.