Difference between current equations of enhancement and depletion mode MOSFET
The physical difference between enhancement and depletion MOSFETs is that enhancement MOSFETs have an additional doping applied to them which increases the threshold voltage \$V_t\$.

So the same formulas do apply only the value of \$V_t\$ is different.

For an enhancement NMOS \$V_t\$ can for example be 0.5 V. That means that when \$V_{gs}\$ = 0 V no drain current can flow.

For a depletion NMOS \$V_t\$ can for example be - 0.5 V. That means that when \$V_{gs}\$ = 0 V a drain current can flow.

To shut off a depletion NMOS a negative \$V_{gs}\$ would be needed.

In practice you will almost never encounter depletion type MOSFETs because they're not so well suited for switching and logic circuits because they're not so easy to switch off. Depletion MOSFETs are sometimes used as RF amplifiers though.