Licensing Implications
In regards to potential licensing issues, unless glaringly obvious (watermarks, do not copy, etc. ), I would not worry about it. I can see several reasons; I will go in to them in greater detail.
Ultimately, if you do not feel comfortable with it, do not do it. Just because something is legally right, does not mean to say it is ethical. If you do not feel comfortable with it, no harm, no foul.
Potential for License Breach
In the event of a breach, I can only see two possibilities:
- The image was originally posted in breach of the concerning license.
- The image was originally posted by the owner, and re-posting it under a 'StackExchange account' constitutes the license breach.
A good rule I tend to follow is to assume good intention. If an image has been posted, I would assume that user has appropriate permission to do so. Furthermore, when we post content to StackExchange, we are stating that we have the right to do so. In fact, when we submit content, we grant ownership to StackExchange1.
According to a moderators reply on a related question, images are no different; and in situations where the user can not grant rights to StackExchange, I would not think they had the appropriate rights to post the image, in the first place. Furthermore, should we be alerted to a breach, all we really need to do is comply with the image removal.
Fair Use
All this aside, there is also the principle of fair use. In most cases, I would assume the image was being used in conjunction with a reasonable interpretation of "fair use" - especially when most of the images I intercept are tailor made, and only include fragments of copyrighted material2 for the point of demonstration. It is important to note that I am not a lawyer; If you wish to know more about fair use, I would direct any questions towards Law.SE.
Context
I think context is important, here. The post you link to is on the photography stack exchange. The replies are all from users who have seasoned reputation in photography, but in comparisson, not much else.
I think it fairly obvious that in the context of photography, image licensing is far more important. Users are asking about photography, and are therefore posting examples of photography. This would be like someone asking about their game, and posting the entire game project. As previously mentioned, the context in which we commonly use images is not as critical, in comparison. We are far less likely to encounter a situation where an image poses a risk of breaking license.
As an additional comment, the answer you specifically link to asserts that '(Moderators)policing copyright; that's not StackExchange's policy.', and issues concerning copyright are a greater issue in protection of our own user's work. While I do not necessarily agree with this, this highlights the relevance of context, in relation to the actual community.
Correcting image links is an appropriate edit; but make sure others know that your making an improvement, too.
At the end of the day, your edits go towards future-proofing the question. These are the sort of edits we want.
That said, it is definitely worth addressing why you made the change, as a comment. Some users simply would not know that we prefer images to be uploaded in a specific way, and why. I personally did not know that correctly uploaded images were uploaded to a 'StackExchange account' until I raised the potential issue of answering a question in the form of a picture.
Other elements to consider, when improving an image.
I will personally make and approve edits that convert the image to the official account. I should point out that a link update is usually not the only improvement that can be made to most image links. Initially (and most of the time, indefinitely), a properly used link should look something like this:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/5Z2Gi.png
Notice the text "enter image description here
"? This is the alternate text. If a user is unable to load the image, the alternate text should take its place. As such, you should include an 'alternate text caption' as an alternate to the image. This caption should clearly convey the same intention as the image, but in text.
You can also include rollover text, by including it in double apostrophes (" "), which is a good way to include sourcing information. I usually include a rollover text to tell the user where the link will take them to, if there is one attached.
If I was editing the question, I would change the above image link to give the following result:
[![Every banner in Terraria][1]][1]
[1]: https://i.stack.imgur.com/5Z2Gi.png "Banners @ terraria.wikia.com"

Final Considerations
- It is worth addressing the fact that you should be trying to improve the entire question. If you make the edit to improve the image, but refrain from reviewing the rest of the question, this might be considered bad form. You should always give the question a browse, and make any other improvements, if applicable. Does the user use correct spelling, punctuation and grammar? Does the title make sense? Are there any other edits you could make to improve the question?
1 It is not as black and white as "you post it, we own it", but there are some fair cases where it is. Answers, for example. For this reason, a user can not 'chuck a tanty' and insist on the deletion of multiple high-quality answers on the basis that they were the author.
2 For example, screenshots that include images of a trademarked editor or other piece of software.