The specific "State Pension" plan you have linked to is provided by the government of the U.K. to workers resident there.
More generally speaking, many countries provide some kind of basic worker's pension (or "social security") to residents. In the United States, it is called (surprise!) "Social Security", and in Canada most of us call ours "Canada Pension Plan". Such pensions are typically funded by payroll deductions distinct & separate from income tax deducted at source. You can learn about the variety of social security programs around the world courtesy of the U.S. Social Security Administration's own survey.
What those and many other government or state pensions have in common, and the term or concept that I think you are looking for, is that they are typically defined benefit type of plans. A defined benefit or DB plan is where there is a promised (or "defined") benefit, i.e. a set lump sum amount (such as with a "cash balance" type of DB plan) or income per year in retirement (more typical).
- Key point: With a DB plan, the organization offering the plan is on the hook to deliver the defined amount(s) at retirement, and is therefore taking on the risk of how the invested funds perform in the meanwhile, because they'll need to make up any shortfall, if any.
(Note: Defined benefit plans are not restricted to be offered by governments only. Many companies also offer DB plans to their employees, but DB plans in the private sector are becoming more rare due to the funding risk inherent in making such a long-term promise to employees.)
Whereas a defined contribution or DC plan is one where employee and/or employer put money into a retirement account, the balance of which is invested in a selection of funds. Then, at retirement the resulting lump sum amount or annual income amounts (if the resulting balance is annuitized) are based on the performance of the investments selected.
That is, with a DC plan, there is no promise of you getting either a set lump sum amount or a set amount of annual income at retirement! The promise was up front, on how much money they would contribute. So, the contributions are defined (often according to a matching contribution scheme), yet the resulting benefit itself is not defined (i.e. promised.)
- Key point: With a DC plan, if investments perform poorly, you get less at retirement. You are taking on the risk of investment underperformance.
DB plans promise you the money (the benefit) you'll get at retirement.
DC plans only promise you the money (the contributions) you get now.