"The ^ character indicates... " or "The character ^ indicates... "

In the first sentence, "character" is being used as the subject noun, and "^" is an adjective further describing which character is being referred to. In the second sentence, "^" is being used as the subject noun, and "character" is an adjective describing what domain of objects the caret comes from.

Both sentences are perfectly valid, and both essentially have the same meaning. However, you may want to choose one over the other depending on whether you want to emphasise characters or the caret specifically.

In particular, if surrounding sentences are about other characters, then it may simplify their logic for you to have made "character" the subject here. If surrounding sentences are about the caret, then the reverse may be true.

In regards to this question, English is the opposite of Croatian.

In your list of examples, the second alternative is the correct one in English. We say "the Wii console", "the Snickers chocolate bar", "the Verizon mobile network", and "the Apple computers". In each case, the capitalized word is an adjective and English puts adjectives before the nouns they modify.

In your real question of whether "The ^ character" or "The character ^" is correct, I would prefer the first alternative. It would also be valid to write:

The ^ indicates the beginning of the string.

At least that seems valid to me, but I'm really a software guy and I'm accustomed to talking about single characters as parts of something bigger. In fact, I would be much more likely to write the following because of its similarity to programming languages where a single character is surrounded by single quote marks:

The '^' indicates the beginning of the string.