@several meanings of hard liG

The hard liG can express wishes —

mA kaz cid duHkha-bhAg bhavet
मा कश्चिद्दुःखॱभाग्भवेत्
"may no one suffer"

It can also be used for blessings and curses, just like the soft liG.

What's the difference between a wish and a blessing?

Blessing affects reality. So if a sorcerer tells you —

bheko bhaveH भेको भवेः "may you turn into a frog"

and you don't turn into a frog, that was a wish. If you do, that was a curse (or a blessing, if you wanted to turn into a frog).

Other uses of the hard liG—

Command:

kaTaG kuryAt
कटं कुर्यात्
"let him make a mat", "get him to make the mat"

Advice:

iha bhavAn AsIta
इह भवानासीत
"Your Honor may sit here"

Invitation:

iha bhavAn bhuJjIta
इह भवान्भुञ्जीत
"please dine here (ie. with us)"

Question:

kim bho vyAkaraNam adhIyIya
किं भो व्याकरणमधीयीय
"should I learn the grammar?"

Request:

mANavakam bhavAn upanayeta
माणवकं भवानुपनयेत
"we humbly request Your Honour to initiate the boy"

Wish:

ciraJ jIver vardhethAz ca
चिरं जीवेर्वर्धेथाश्च
"live long and prosper"

Prayer:

bhavati me prArthanA vyAkaraNam adhIyIya
भवति मे प्रार्थना व्याकरणमधीयीय
"I have a prayer that I may learn the grammar"

By rule loTca, the /loT tense too can express these senses, and by praiSAtisarg..., so can the /kRtya affixes.

The original sUtra just says "/liG", why do you translate "hard liG"?

This rule does not apply when the /liG is soft: because of the exception liGAziSi, the soft liG can only mean a curse or a blessing.