I have been having the same issue and trying all kinds of approaches to solve this.
At the moment I can only share what has NOT been helpfull.

Case:

I do lots of outreach to newspapers, blogs, partners and set up cooperations and links.
To one that would give me most benefits is one that I did towards a German newspaper concern. The did wrote an article and linked towards my boatcharter page.

These are well established newspapers with the following DR:
DR 53 - DR 60- DR 43 - DR 42 - DR 58- DR 67 - DR 74

Ahref , SmallSEO tools, semrush and Moz all show these links accept Google does not....

Tried solutions:

1. Pinging
   I have tried pinging all these pages via ping o matic, prepost seo and pingler to see if google then would pick it up. After waiting 7 days nothing

2. Crawling my website
   As my website is google indexed and when I make a new blogpage I upload the link directlty in the google search console for fast indexing.

I made a thank you blog page containing links to the websites that contain my backlinks, So that when google crawls my page it should see these links and from there go check on those pages.

Waited 2 weeks and also did not work...

At the moment I am thinking out a new strategy as in total I still have about 15 real DR 50+ websites linking to my small website and it would be a considerable help for my DR.

Solution:
If anyone else has these issues, try to strategy that I did and leave your findings here.

@thepeterlunt Do at least some basic keyword research, later you can always change it when you have them more pinpointed. In the beginning I was mostly using the ones that I would use as searchterms. But even in English language if you have somebody from australia or from scotland they will use different english sentence structures even.

It took me 6 months to get a good list of keywords for my SEO

@BlueprintMarketing said in Backlinks on Moz not on Google Search Console:

Google's John Mueller was answering questions around an old topic of Google showing only a sample of links in the link report in Google Search Console on Reddit. He said that generally, if the links are not shown in the tool, then they are "pretty irrelevant overall."

The part "pretty irrelevant overall." is surprising. In my case I have links from German newspapers DR 60 , 43,42,58,67,74 to my website.
There seems to be something else playing as news websites make sure theyre articles and pages get crawled all the time.
I used Pingler, ping-omatic and prepostseo and pinged the Newswebsites that relate to my article. As well I made a [BS blogpage](link url) for a couple days where I link to all relevant websites where I have backlinks.
At the moment I am waiting the see the results and will come back here to share.