Hello
yes, the Lusitania was torpedoed "without warning", as it did not use the corridor for passenger ships, it was camouflage-painted grey (the paintings of the sinking Lusitania picturing a black, white and yellow ocean liner are propagada, again) and had ammunition and troops aboard. Following a zigzag course (which was not allowed for neutral and hospital ships), it was misidentified as a troop ship, and sunk. Only its last zag brought the L. before the tubes of the U-boat, which would have stood no chance against a ship running at 20+ knots.
News from the long-closed british admiralty archives make it seem possible that Churchill had intended to use the "Lusitania" to lure the US into war at England's side, in the british admiralty not warning the captain of the L., in spite of knowing the exact position of the U-boat being in the L's direct path which then sunk her.
The US were on the brink of declaring war to England, due to the US had "guaranteed free seas for all" in the early war, and Wilson was furious about the british blockade, and the famine in Germany. Remember there were a lot of germans and people of german ancestry, in the US. When the one german trade U-boat rendered the blockade useless, there was an outcry in England in that the US should be forbidden to trade with Germany.
As you rightly said passengers were warned in US newspapers to board the "Lusitania" because of its well-known "secret" of being an auxiliary cruiser run by the british navy by its original papers (it had been a trick by the company for getting money from the government to build it, but at that time this could only be known in the inner circles - to all else and Lloyd's the L. was a military ship, and certainly used as such). Some of the US passengers even complained that the L. obviously carried troops and war material, thus threatening their lives.
British freighters, tankers of all kinds and hospital ships were often (ab)used as being "neutral" in wrongly running false flags during WW1, even using hospital ships as ammunition and troop transports, in the mediterranean.
German hospital ships anchored before Kiel and other coastal cities were sunk in the baltic sea, by british "submarines", no reluctance here but seldomly mentioned. (According to Churchill the good ones were the "submarines", while the bad buddies were to be called "U-boats"). The propaganda war is indeed one of the less "heroic" efforts, of WW1.
The "unrestricted" U-boat war of Germany was "unrestricted" only for a few months (b.t.w. an unrestricted war, be it U-boats/submarines or surface units, was led by England right from the beginning).
Kaiser William 2nd failed to communicate this clearly to the US against british propaganda, because he feared the reaction of the german population, being angry about the british declaration of war (why had they?). After the famine in the hunger winter killing 750.000 civilians due to the british blockade (Germany was dependent on sea trade as much as England was), the population expected an unrestricted war against the agressor as they saw it, which was not led (!); but telling this officially would have isolated William 2nd even more, for being too pro-british.
Even during the two months of unrestricted U-boat war by Germany, most ships were still hailed and stopped, since most U-boat commanders refused to sink ships without warning, according to international treaties - even if they were not being followed by England. The almost only exception were the smaller boats of the Flanders flotilla with its tiny coastal boats, not being fast enough to chase even the slowest freighters - a hailed and warned ship ordered to stop, would just open up steam and run away.
The thing is, german U-boats sunk more ships in a month, that they did in the two of the "unrestricted" war.
So if you talk about the "Lusitania" and repeat the propaganda of the time, you might as well mention the "King Stephen" with the L-19 airship, and the Q-ship "Baralong" incident.
Greetings,
Catfish