word choice - Using "use" with "to" and "for" when expressing purpose

I recently wondered about the use of "to use" and other verbs when expressing the purpose of an action.
I noticed that purpose is often expressed by having a verb followed by "for" and a progressive form, as in the following example:

• For memorizing her shopping list, my grandma started using a cell phone.

Alternatively, this seemed acceptable, too:

• (In order) To memorize her shopping list, my grandma started using a cell phone.

However, I recently received corrections for a text in which I frequently used to former version in conjunction with the word "to use".
This was changed by the corrector every single time, and I do not quite understand the reason for that.

Consider the following sentences:

• I used a short example to explain my main ideas.


• I used a short example for explaining my main ideas.

Is one of the two phrases incorrect or somehow discouraged? If so: Why is that, and what is the difference between "to use" and other verbs?

NOTE:

The example above may be more clear when switching to passive phrases:

• A short example was used to explain the main ideas.


• A short example was used for explaining the main ideas.

I think having "was used to" in that sentence makes it sound weird. That is why I preferred the version with "for" until now.