The first one sounds more neutral to me, just mentally add a questionmark or a "right?" to it. They're leaving space for you to respond if the statement is true or not. This is kinda like how conversations go, people leave space for the others to chime in and take their part in it.
If someone "claims" something happened, it seems like an accusation or interrogation.
The main takeaway is that everyone interprets things differently and prefer different styles of communication. Just keep in mind that your interpretation is not always the same as what is being intended.
I don't know about thin skinned... it depends on a bunch of factors aside from mere words and pauses, written down here. Do you generally tend towards the defensive (feeling attacked/accused/wronged in conversation, and in turn faulting the other person for feeling that way) when dealing with other people or in certain settings?