more often than not they point to individuals who are Polish and proclaim them to have been Lithuanian patriots.
That is true only from the contemporary point of view. And this is true when one assumes that their Polish language would for them be the only criterion to feel entirely Polish as opposed to the fact of feeling entirely Lithuanian (meaning the Grand Duchy of Lithuania).
Since when saying that from the contemporary Polish point of view the fact that they are Lithuanians patriots is exceedingly amusing, it only reveals the ignorance of the contemporary Poles of the history of their country which they shared with other nations. Adam Mickiewicz or Tadeusz Kościuszko will call themselves Lithuanian patriots with no hesitation whatsoever, and every page of the book "Pan Tadeusz" by Adam Mickiewicz is proof of that. The problem for most of the contemporary Poles, but not for the Poles of the past, is that those outstanding Lithuanians could see no opposition between "Polishness" and "Lithuanianness", treating the Grand Duchy as part of a wider cultural and political entity. And if this is so, we, the Poles, have no right to claim them as only our own. Do the English have the right to claim those Scottish or Irish who use English in every day life as English, since most of them do not know Scottish or Irish? No, but they do have the right to claim them as British. In reality, the term "Polishness" as used commonly in the Commonwealth of Poland and Lithuania until the end of the 18th century can be compared to the term "Britishness" in its modern sense. If only our ancestors could have remplaced "Polish" with a more universal term in the past, the above discussion would have no sense. But in spite of that, they would start to slowly use the term "Polish" to describe the whole entity, with the term "Lithuania" and "the Crown" evolving to describe the two separate parts which was making it up.
However, the difficulty in using the term "Polish" for the whole political entity of the Commonwealth still presented a problem for the official state nomenclature. It could not be borrowed in the forms described above since the head of state's title was "King of Poland" and "Grand Duke of Lithuania", with both states which formed the Commonwealth formally being called accordingly. In my view, if the Commonwealth had not been erased from the map of Europe in 1795, the term "Polish" in its common 18-th century meaning would have stood a great chance to develop over time into a commonly used term of "Both Nations" (Obojga Narodów). The 3rd of May Constitution of 1791 seemed to have opened the path for it; despite of the abandonment of the existing political division of the Commonwealth into the two states (or perhaps beacuase of that!), the Constitution gave birth to central ministries which were now being called "Ministries of Both Nations", for example the counterpart of the Home Office was given the name of "Komisja Policji Obojga Narodów".
Another, third part of the Union was obviously lacking, but it was introduced to the official emblem of the Commonwealth-to-be-recovered during the times of the January Uprising of 1863, where the Archangel Michael of Ruthenia (Ukraine) accompanied the Pahonia of the Grand Duchy and the White Eagle of the Kingdom of Poland.