You say you are substituting the juices for milk. You have to ask yourself, what roles does milk play in the recipe, and how well will the juices in question fill those roles?

What does milk contribute?

The milk adds:

• Liquid, to gelatinize the starch in the flour to create the structure of the cake; the water will also contribute to some gluten formation, but in most cake techniques, this is minimized
• Very minor acid component
• A small amount of fat, but compared to the oil in the above recipe this becomes negligible
• A small amount of sugar, but compared to the sugar in the above recipe, also negligible
• Some proteins and other milk solids which will interfere slightly with gluten formation, again enhancing tenderness; they also help to retain moisture so the cake stays fresh longer
• A very milk neutral flavor that comes out as a general contribution to the richness

The fruit juice

In contrast, what do fruit juices bring to the recipe? I am ignoring the heavy/light or thin/thick distinction because it means very little.

• Water--exactly the same as the milk
• Significant acid. Fruit juices tend to be much more acidic than milk. You might wish to experiment with reducing the baking powder, and adding some baking soda to use the additional acid to power the chemical leavening, while neutralizing some of the overall acidity
• Most fruit juice is essentially fat free, but as the fat contribution from milk is negligible, this is not an important distinction
• Possibly some pectin or other long polysaccarides which will contribute to the overall structure and mouth feel of the cake, but this effect is probably negligible
• Flavor. The fruit juices will have a strong and significant flavor--of course, this may be a positive, but it will influence the outcome of the cake
• Sugar. Fruit juices tend to be sweeter than milk, so the overall cake will be slightly sweeter. This is probably somewhat mitigated by the increased acidity, and again, compared to the overall sugar in the recipe, probably not a large contributing factor.
• Fruit solids (proteins, minerals and so on). Much like the milk solids, probably contribute somewhat to the more tender crumb, but hard to characterize.

The truth is, the "heaviness" or "thickness" (by which I infer you mean viscosity, or how fast the juice flows or pours--or how easily it coats a spoon, which is an easier way to see it) is not that important.

You could use pureed fruit with almost the same result--for example, canned or pureed pumpkin or mashed banana. Even pureed fruit is still mostly water.

Still, every juice or fruit puree is its own entity. I would keep a little diary of how each trial worked out, whether you like the results, and adjust the next try accordingly.

Edit:

What juices to use?

I realized I never specifically answered the question, what juices to use?

Use whatever juice (or fruit puree) you like, based whether you like the flavor and how the cake turns out. Each time you try something new, there is some risk, but you should soon find ones you like, and which perform well.

Two things come to mind 1. Do you peel your carrots? 2. Metal pan reaction . Try peeling your carrots. I have gone as far as cleaning my fresh carrots with a green scrub pad. But have found nothing works as well as peeling them. Even if you never peeled them before, when you said you did not have this problem, It could be a reaction to something thats being "added" to the carrots now days. The other thing I had a problem with in the past was the type of pan I used. I found that some metal pans would discolor my cake. And the longer the cake is left in the pan it would effect the flavor as well. Hope this helps someone. Have a great day!