Schmitt Trigger Functionality Not Working, Questions
Just plug those resistor values into the calculator you linked in your question and look what happens: -



The thresholds are switching between about 4.2 volts and 4.6 volts - that's the first thing - your input needs to be able to cross those boundaries for your circuit to switch.

Next, take a look at the LM358 data sheet and you'll see that input voltages are only valid between 0V and 3.5V (on a 5V supply) - don't expect this to do what you want when the input is above 3.5 volts.

Next take a look on the d-sheet to see what the output can do on a 5V rail, The d-sheet implies the highest output voltage that it can muster is typically 2V below the positive rail i.e. 3V.

My estimation is that your circuit is kind of working but maybe with the invalid 5V input in scenario 1, you get a weird inversion thing going on (it happens a lot on op-amps).

Welcome to the imperfect world of real op-amps!!

Try lowering thresholds to around 2V and 2.5V and see what happens when the input rises above 2.6V - I'd expect the output to swing down to 0V. When the input drops below 1.9V, I'd expect to see the output swing above 3V.