Sounds like you mostly understand it, but here a couple more things to know:

Pensions usually require you to work at the employer a minimum number of years to get the monthly benefit. It varies by plan but can be rather long for a government pension. If you leave before then, you should be able to get back the amount that is considered your contribution, but not any of the employer contribution.

You definitely can open your own IRA in addition to the mandatory workplace plan. The workplace might also have another retirement savings plan that you could use in addition to the mandatory one.