PPer Cambridge Dictionary Online,
commit verb (PROMISE)
[I or T] to promise or give your loyalty, time, or money to a particular principle, person, or plan of action:
Like so many men, he has problems committing himself to a relationship.
The government must commit itself to improving healthcare.
Question is, would it be grammatically acceptable to rephrase the example sentences above as follows, while keeping the exact same meaning?
Like so many men, he has problems committing to a relationship.
-and-
The government must commit to improving healthcare.
The government must commit itself to improve healthcare.
The government must commit to improve healthcare.
EDIT:
Please, consider this other example:
We commit ourselves to providing [...]
We commit to providing [...]
We commit ourselves to provide [...]
We commit to provide [...]