It's not so much a matter of what words you use, as keeping distinct the several different things that are going on: Doppler effect, light travel time delay, and time dilation. "Warping of the passage of time one frame relative to another" is an OK description of time dilation, but only *after* you've corrected for the Doppler effect and light travel time delay; you can't directly observe time dilation in this sense.
Here I was using "appear" to mean "the way the incoming light actually appears to you directly", which is primarily driven by a combination of the Doppler effect and light travel time delay. You are moving towards things in front of you, so the light signals from them will appear speeded up, because of the Doppler blueshift plus the fact that as you get closer, the light travel time delay decreases. You are moving away from things behind you, so the light signals from them will appear slowed down, because of the Doppler redshift plus the fact that as you get farther away, the light travel time delay increases.
You can *calculate* that, after you've corrected for those effects, both the things in front of you and the things behind you are time dilated compared to you--they are aging more *slowly* than you are, as seen in your rest frame. See below for further comments on this.
In an actual situation, yes, this would likely happen. In discussions like this, though, we can assume that we can ignore this effect; basically we are assuming that there is always enough radiation coming from the object that will get redshifted or blueshifted into a range we can detect, so we can get enough information to construct what you call a "real representation of what is occurring".
This brings up another good point, which I briefly referred to above. "Time dilation" by itself is not enough to tell you what some other object's clock will read when you reach it. For example, suppose I fly at a speed very close to the speed of light from Earth to some galaxy a billion light years away. Suppose I know, somehow, that clocks on my destination planet in that galaxy are exactly synchronized with Earth clocks (and suppose also that the planet is exactly at rest relative to Earth). Then I expect that, if I leave Earth at time t = 0 by Earth clocks, I will arrive at t = 1 billion years (plus a little bit because I'm not quite traveling at the speed of light) by the destination planet's clocks, whereas a much shorter time (say a year) will have elapsed on my clock (and I will only have aged a year, etc.).
*But*, if I look at things from my rest frame while I am traveling, it will seem to me that clocks on Earth *and* clocks on the destination planet are running much *slower* than my clocks are. That is, once I correct for the Doppler effect and light travel time delay, I can calculate from the light signals I receive from Earth and the destination planet that they will only age by a fraction of a second during my entire trip, while I age by a year. So it seems at first glance that I should predict that clocks on the destination planet will read only t = 0 + a fraction of a second when I arrive. Why, then, do I find that they actually read a t = a billion years plus a bit?
The missing piece here is called "relativity of simultaneity". In the rest frame of Earth and the destination planet, the event of my launch from Earth happens a billion years (plus a bit) before the event of my arrival at the destination planet. That means that my launch from Earth is simultaneous (in the Earth-planet rest frame) with an event on the destination planet that happens a billion years (plus a bit) before I arrive. Call this event (when clocks on the destination planet, synchronized with Earth clocks, read t = 0) event P.
However, in my rest frame while I am traveling, those two events (my launch and my arrival) are only a year apart (because that's how much time I experience during the trip). That means that, in my rest frame while I am traveling, the event of my launch from Earth is simultaneous with some *other* event on the destination planet, call it event P'. When I calculate that only a fraction of a second elapses on clocks on Earth and the destination planet during my trip, what I am really calculating is the time, on the destination planet's clocks, between event P' and the event of my arrival. But event P' doesn't happen at time t = 0 by the destination planet's clocks; it happens at t = 1 billion years, plus a bit, minus a fraction of a second.
In other words, to predict what the destination planet's clocks will read when I arrive, it's not enough just to know about time dilation; I also have to know how the destination planet's clocks are synchronized, i.e., when their "zero" of time is. If their "zero" of time occurs when I am spatially separated from them (as it does for the destination planet--when their clocks read t = 0, I am on Earth, just launching), I have to take that into account as well as time dilation in order to know what their clocks will read.
As I hope I've shown above, it's more complicated than that.
This is a good question; let's look at what things look like from both Earth and the destination planet.
If we're on Earth watching the trip, we see the ship leave at time t = 0. Since its destination is a billion light years away, we will see the ship's arrival at time t = 2 billion years plus a bit (1 billion years plus a bit for the ship to get there, plus 1 billion years for the light to get back to us). So the whole trip will appear "slowed down"; it will take about twice as long for us to see the trip happen, as it takes for the trip to actually happen. That doesn't double the actual travel time: the ship still arrives in 1 billion years plus a bit. It just takes twice as long for us to receive all the light signals emitted from the ship during the trip.
If we're on the destination planet watching the trip, it's more interesting. Since the launch point (Earth) is a billion light years away, it will take a billion years for the light from the launch to reach us. So we will see the launch at t = 1 billion years, and the ship will actually arrive very soon after, at t = 1 billion years plus a bit. In between those two events, we will see *all* the light signals emitted by the ship during the trip, drastically speeded up.
Note that in both cases, we can correct for the Doppler effect and light travel time delay to calculate that the ship and its crew only age by 1 year during the trip. So what we actually see can be quite different from what we calculate that the ship and its crew will experience. That's why it's so important to keep the different things that are happening distinct.