@mechpolaris
The relationship happened before being awarded the title of crown prince, and even as a duke they still hid it from the mc.
Actually at some point she mentioned that she brought them closer, so you can neither be sure nor does it change the essence of the problem. If anything his cousin and the baron got in a relationship thanks to her. Even as a baron he is still expected to produce an heir, being a crowned prince just solidifies this further. I don't think anyone will argue that it is reasonable for him to publicize the fact he is gay considering the time period and his social standing.
Anyway, again is it not considered as cheating when they are already engaged or to be wed, an affair was hidden and still ongoing without the partner's knowledge?
Not in the usual sense of the word, no. This is because their engagement is in name only, and there's no actual romantic relationship to speak of - he's not technically her "partner" in most senses of the word. It's like getting a boyfriend/girlfriend, then not doing any of the "activities" that come along with the territory, and believing that you are actually in a relationship. By all means this is not much more of a relationship than having one-sided feelings towards someone (and we have a good reason to believe her feelings are an unhealthy obsession with the baron, rather than love, as evidenced by her actions and them not engaging in any romantic activities).
is this still not considered cheating?
Depends on the culture and timeframe. The issue is that the current western meaning of the word has a very different connotation than what we saw here. So yes, you can technically call it cheating, but it just doesn't hold the same meaning as what's commonly understood by that. You can see how that may be an issue, especially when one blames the baron for "cheating" while using the current sense, to refer to what was in the manga, and use that as an argument.
To quote wikipedia (referencing a research paper on that topic):
What constitutes an act of infidelity depends upon the exclusivity expectations within the relationship.
Considering there are no romantic expectations or a relationship to speak of, I do not believe we can speak of exclusivity expectations pertaining to love (not that they have even discussed that). And as far as his duty with regards to marriage and child rearing goes, we have a good reason to believe that he was planning to do well by it. So if anything he went as far as possible, while being gay, to honour his formal engagement. I am not sure what more you wanted from him in that regard - to become heterosexual magically? I am sure that if he was, he would have loved her, considering he loves the cousin that looks like her. The problem in that case would have been that she would have probably disregarded him because he would have been deemed "disgusting" for looking at her. It's a "self-fulfilling tragedy".
in a relationship others can be included and still not be cheating
If you want to go with this, you need convincing arguments that this is indeed a relationship, and not just a politically arranged contract. So far, from the information of the story you cannot provide such arguments.
the partner is aware and willing to share the guy to not be considered as cheating
It takes at least two people for a relationship and for an agreement - and an actual discussion of the matter - which is not the same as: "I heard you can summon beasts". She never once questioned what the duties and rules in that "relationship" are, since from the very beginning she knew that she got close to him by using his greed for power and achievements. It's pretty obvious that the idea of fidelity of the baron (marrying her and getting her pregnant) do not align with her ideas (unconditional love) - where the problem comes from. But from all of the actions and events it should be clear that their relationship is supposed to be closer to an arranged engagement, and in that sense the baron's ideas are a lot more adequate. So people trying to blame him for "cheating" are clearly biased if they willingly disregard the role she played in this and the nature of the "relationship".
even using the opportunity to use her death for a child to raise together
He could have nullified their engagement at any point and found a different spouse that would be understanding of his "preferences" and didn't expect love from him. He didn't because I can only assume he wanted to honour his engagement with the MC, despite the fact that it would have most likely tarnished his reputation (considering what people thought of her). So I do not think that your assesment here is correct. He could have had any woman bear his child considering he's the crowned prince - he was planning to choose our disfigured MC even then.
