The first site you link to is a pretty good summary.
Without wishing to criticise the quality of their software, the second site you link to is an incoherent ramble. It says
In particular, if I modify the code within my company, am I obliged to make the source code available to internal users (employees of my company)? We wish to remove this ambiguity: for us NO.
The GPL traditionally doesn't regard distribution within a company as conveyance, but that refers to activites that engage ss 4-6 (GPL and AGPL). The extra language of AGPL, in s13, says merely
your modified version must prominently offer all users interacting with it remotely through a computer network (if your version supports such interaction) an opportunity to receive the Corresponding Source of your version
The language I have highlighted says nothing of "conveyance", and gives no reason to think that it should be interpreted with the same traditional rules and exceptions. Certainly the authors of ProjeQtOr, who are presumably the rightsholders in the whole of their code, can distribute on any terms they like. But they cannot modify the AGPL as they have tried to do, and still call it the AGPL (that's forbidden by line one of the AGPL!). They also can't, as I read it, do what they want with a s7 permission, since s13 starts "Notwithstanding any other provision of this License.
Essentially, they can either distribute under AGPLv3, or under a weird crayon licence that looks a lot like AGPLv3 but exempts intra-company interactions from s13. What they can't do is the latter while still correctly claiming to be doing the former.
Their table of AGPL obligations, which says that if you "change ProjeQtOr to propose hosted services in SaaS mode ... You have to distribute your version under GPL license" directly contradicts AGPLv3 s5b. I can understand how you might be confused after reading this piece. I recommend ignoring it.
Your third linked site is an opinion piece focussing on whether the AGPL is good or bad, which has nothing to do with the rights and obligations it grants you. I won't comment on it here.
In answer to the question in the title: you must share such modified software - and not just your modifications, but the complete corresponding source - with anyone to whom you distribute any portion of the software, and with anyone who interacts with your modified version over a computer network.
