Whether you have a pension or a 401(k) has a very big impact in how you organize your career in the U.S. If you have a pension, you're strongly incentivized to stay with one employer for a lifetime, because you lose it if you leave and your benefit usually depends upon length of service. That in turn means that you don't care how your resume looks to the outside world, you don't invest as much time in professional development, and you generally accept whatever advancement schedule is available in the organization.
If you have a 401(k), you're strongly incentivized to maximize your earning potential, because you only get out of it what you put into it (plus investment returns). That usually means switching jobs, which means you need to consider how your professional experiences will look to the outside world. It means that you need to be constantly up-to-date with how the industry is changing around you, and with how the financial markets are doing, and manage your own finances & career development. On the plus side, you take your 401(k) with you if you quit or move, and so you have greater flexibility. And you're not exposed to the risk that your employer could just die and take your pension with it. You are exposed to the risk of the stock market crashing, though.
Because life strategies are so different based on the different retirement schemes, people will assume certain things about you if you say you have a pension, and they'll assume other things if you talk about your 401(k). If you say you have a pension when it's really a 401(k), those assumptions are likely wrong, which leads to awkwardness all around. That's why they're different terms in colloquial usage.
Typically no, unless you're unvested, I think. You may get a much smaller benefit, though.