moan vs groan

moan a. A low, sustained, mournful cry, usually indicative of sorrow or pain.

groan :

In your definitions, "sustained" is unique to "moan."  A moan would be made by a living thing, or by something which imitates a living thing.  It would tend to be a vowel, or singing sound, although probably not a constant pitch. 

Hopefully Jim or Kooyeen will get in on this.  They're really into sounds.

For groan, your #2 definition describes non-living sounds, in the same category as "creaking."

The old house groaned under the force of the hurricane.

As you can see from your definitions, both terms can indicate pain, suffering, and unhappiness; so it's impossible for the context to pin these cases down.  You specify which sound it is, and hope the reader understands the difference.

We heard a moan/groan coming from behind the bushes, and discovered a man who had been badly beaten.

The woman held the dead baby in her arms, moaning softly.

  - A.

moan 

 1 : LAMENTATION, COMPLAINT <made a great moan if he had to work -- D.H.Lawrence>
2 a : a low prolonged sound indicative of pain or of grief b : any similar low mournful or murmuring sound

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groan

1 a : to make a deep usually inarticulate and involuntary often strangled sound typically abruptly begun and ended and usually indicative of pain or grief or tension or desire or sometimes disapproval or annoyance <the dying man groaned with every jolt of the ambulance> <men groaning under the weight of the loads they were carrying> <groaned with rage and frustration> b : to make a harsh sound (as of heavy creaking, grating, rasping) upon subjection to sudden or prolonged strain (as of a heavy load) <let himself fall into an armchair which groaned under him -- H.M.Ledig-Rowohlt> <wagons that swayed and groaned up the hill>

2 a : to experience pain or grief enough to make one groan <groaned when he read the telegram> b : to undergo strain or oppression or overburdening enough to make one groan <men who were once free now groan in slavery> <tables which groaned under the weight of good things -- Norman Douglas> c : to desire something intensely enough to make one groan <groaning to be with her again> <death groaning for fresh victims> d : to disapprove of something or become annoyed with something enough to make one groan <groaning over their stupidity> <one groans at the absence of an index -- Sean O'Faolain>

 http://unabridged.merriam-webster.com

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New2grammar

Avangi, you're going to hate me for pointing this out.

If you're speaking of the moan-slash-groan, that's actually my second example.  I specified your #2 definition of groan as applying to non-living things.

I'll read it again.

Best wishes,  - A.

Edit.  "Groaning" would be fine in my (last/third) example.  I was hard-pressed to come up with a case where groaning couldn't be substituted for moaning.  I can see that I failed miserably.

Also, I meant to imply that a groan is by contrast, NOT a vowel, or singing sound, in addition to not being sustained.  I'd call it more of a noise.