The OP is about the invented story that the two small Russian squadrons that deployed to the US in 1863 were there to show support to the US. They weren't.
The Russian Navy had been effectively destroyed in the 1853-6 war. The allied navies spent 1855 cruising the Sea of Azov destroying every single military facility and burning the food supply for the Russian army at Sebastapol. The newly developed littoral capabilities the RN and MN developed in 1855 were terrifying.
The 1853-6 war had completely destroyed the Black Sea Fleet, and the Black Sea clauses of the Treaty of Paris prevented Russian from reestablishing it, or fortifying the Black Sea ports. Russia did not repudiate these clauses until the Franco-Prussian war distracted everyone. The same treaty also forbid the fortification of Åland. The effect was to completely neuter the Russian navy.
In 1863 there was a serious concern by the Russians that the Crimean War was going to reignite. The "three powers" (UK, France and Austria) delivered joint notes to Russia regarding the Polish uprising, but all three had different objectives. The Polish rebels killed any hope of support by declaring that
Galicia was to be part of their new state, taken from Austria. This killed Austrian interest in supporting the Poles, and the British thereafter simply noted that Russia should not attempt to alter the Treaty of Vienna (by annexing Poland into Russia proper) and should emancipate the Polish serfs (as the UK had forced Russia to emancipate the Russian serfs). The RN then sent the Channel fleet (including three ironclad frigates;
Defence,
Resistance and
Royal Oak and three steam ships of the line;
Revenge,
St. George and
Trafalgar) to cruise the Baltic and indicated the price for non-compliance was that Britain would carve off a separate Polish Kingdom (not the declared republic).
It was against this background that the Russians sent out a dozen cruisers to go to the US in case the British followed through. Russia indicated compliance with British wishes; they did not formally annex Poland, and emancipated the serfs in January 1864. This ended British interest in intervention, and the French were not powerful enough to intervene without Britain.
Russian histories I've read seem to be different, ignoring the fact that they in fact complied with British demands.