Diode circuit using real model
There are three, not four, circumstances to worry about. Assuming that both diodes are ON, you have the left-side case below. It should also be absolutely clear to you why the middle case loses the particular branch of the left-side case that it does lose. And finally, the right-side case should be similarly obvious.







simulate this circuit &ndash; Schematic created using CircuitLab

Clearly, the left-side case only works if you find the circumstance for \$V_I\$ that achieves the indicated \$V_X\ge +2.1\:\text{V}\$.

The same will be true for the middle case, only now you need to find the circumstance for \$V_I\$ that achieves the indicated \$+600\:\text{mV}\le V_X\lt +2.1\:\text{V}\$.

The right-side is too obvious to bother saying much about.

You should also find that these three models are consistent with each other (you should find that the results make sense to you.)

At this point you may be able to recast your original schematic now that you've worked out when current is flowing in each branch to work out \$V_\text{O}\$ on your own.