This is going to be a confusing answer. So let me label the dimensions with a different name, which will help when you read the rest of the answer.
- ND - New dimension, where Rick and Morty jump to.
- OD - Old dimension, the one that gets Cronenberged.
The question now becomes: which of these dimensions is C-137?
ND = C-137?
This seems the most likely, for a multitude of reasons:
- The show has never mentioned dimensions or any of their names until after R&M have already jumped from OD to ND.
- We first hear the name "C-137" from the Council of Ricks (in Close Rickcounters of the Rick Kind). Notice how it happens:
- This takes place in ND.
- The family is eating together.
- The Council portals in, interrupt dinner, and immediately addresses Rick as "C-137". Their first line starts with "Rick Sanchez of Earth Dimension C-137 ..."
- Although not impossible, the Council has never shown to be aware that Rick is from another dimension (OD) than the one he is in (ND). It's highly likely that the Council was referring to Rick by the dimension that he's currently inhabiting.
- You linked to an image of Rick filling in a Jerryboree form. On the form, Rick lists his own dimension as C-137. This happens at a point in time where Rick is pretending to be ND Rick.
But why does Rick fill in "N/A" for Jerry dimension? This doesn't make sense.
Actually, it does make sense. Asking for Jerry's dimension after already having asked for Rick's dimension (on the form) would only be meaningful to fill in if this Jerry is from another dimension as this Rick.
If Rick fills in "N/A", that means it's not applicable to make this distinction, which means that Rick is implicitly saying on the form that him and his Jerry are from the same dimension.
This also makes sense. Rick and Morty are hiding the fact that they are from OD, and they consistently act as if they are from ND. It makes sense for Rick to put his assumed identity (ND, C-137) on a Jerryboree form.
(I would hazard a guess that the Council of Ricks members also make use of the Jerryboree.)
OD = C-137?
There is little evidence that suggests OD must be C-137.
This is further compounded by the fact that we never hear the name "C-137" while our Rick and Morty are still in OD.
And then there's the fact that the Council of Ricks referred to our Rick as "Rick Sanchez of Earth Dimension C-137". If C-137 is actually OD, that would mean that the Council was aware that this Rick is the Rick from OD (and not from ND, where they came to get him).
The Council couldn't have known that our Rick is from OD.
Later, when they suspect our Rick of being guilty of the murders, the Council of Ricks wants to scan Rick's portal gun to ascertain where he has been:
Some Rick - Scan his portal gun!
Our Rick - Oh, come on. Don't look at another man's portal-gun history. We -- we all go to weird places.
Some Rick - Yes, but it appears you alone have been going to the exact timelines and locations in which the murders occurred.
This directly proves that the Council of Ricks needs to access the portal gun locally before they know where the portal gun has been. This means that they cannot be aware of Rick's travel history before they took him to the Citadel.
This means that when the Council of Ricks entered the scene and addressed our Rick as "Rick Sanchez of Earth Dimension C-137", they must have been unaware that Rick has travelled (as they have not yet scanned his portal gun), which further suggests that they are therefore referring to Rick by the current dimension's name.
It's never explicitly confirmed, but it's highly likely that C-137 is the new dimension that Rick and Morty jump to after Cronenberging their home dimension.