pronounciation question?

Your understanding is absolutely right.

In some English dialects, the "g" is sounded at the end of "-ing"; in some other English and American dialects, the "g" is omitted altogether, and "-ing" is pronounced "-in". Both these variations, however, sound and are considered uneducated.

I think there are 2 kinds of "ing" sound you might be talking about.
First, in the case of verb -ing forms (gerunds), as in:
walking, doing, eating,
etc., it's common to drop the g when speaking. So these sound like,
walkin, doin, eatin.
But this only applies to -ing verbs.

But from your description, I think you might be talking about the "ing" sound in general, as in "sing," "king," "thing," etc.
In this case, the g is not quite silent, but it's not a normal "g" sound like in "go." It's a soft, slightly nasal sound that can be hard to hear for English learners, depending on your first language.

Your description of tongue & mouth shape is correct. Saying "king" is just like saying "kin," except at the end, your tongue moves into a "g" position instead of an "n" position

It's not bad grammar to let the "g" sound drop off of "-ing." At least, when you're speaking with an American accent, it typically isn't. There are occasions when it will and does sound uneducated, as when people over-emphasize the -in' sound.

But to answer your original question, the "g" is sometimes silent and sometimes not. It just depends really on how you talk and what the following letter is.

I wish I could be more specific. Could you perhaps tell us where you're from...that might help us tailor our answers and pronunciation can be perceived differently in various countries (my POV now is as an American).

***Edit: The first poster said that letting the "g" drop off is considered uneducated. Well, I'm not trying to push, but it really depends on the situation. Most Americans don't pronounce the "g" as it should be, but instead make the sound a closed, nasal. If you know IPA symbols, I can show you the morphed sound we make :)