I finally solved the issue. I don't know, till now, why user was not visible in User Information List in the browser but was visible in the json returned by the rest api request, but using the EnsureUser method solved the problem. This is the script that I've used :
function ensureUser(webUrl,loginName)
{
var payload = { 'logonName': loginName };
return $.ajax({
url: webUrl + "/_api/web/ensureuser",
type: "POST",
contentType: "application/json;odata=verbose",
data: JSON.stringify(payload),
headers: {
"X-RequestDigest": $("#__REQUESTDIGEST").val(),
"accept": "application/json;odata=verbose"
},
success: function(){alert('success');},
failure: function(){alert('failure');}
});
}
var loginName = '<LoginName>'
ensureUser(_spPageContextInfo.webAbsoluteUrl,loginName);
After that, the user was visible again on the User Information List browser UI and we could add him to the people picker and find him after editing the same item.