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Regarding your trouble understanding what happens when you type sage in a terminal:

run which sage to try to figure out what gets run when you type sage
run echo $PATH to check what other locations are searched before /usr/local/bin
check if you defined an alias in your .bashrc or .bash_profile or .bash_aliases
if the ln -s command you ran did not succeed, try with

sudo ln -sf /SageMath8.1/sage /usr/local/bin

Regarding your trouble understanding what happens when you type sage in a terminal:

run which sage to try to figure out what gets run when you type sage
run echo $PATH to check what other locations are searched before /usr/local/bin
check if you defined an alias in your .bashrc or .bash_profile or .bash_aliases (see this blog post about .bashrc and others)
if the ln -s command you ran did not succeed, it might be either because there already is such an alias there, or because of a permissions problem; try with

~~sudo ln -sf /SageMath8.1/sage /usr/local/bin/usr/local/bin~~