There are two real questions here. The first is what are you liable for. The second is whether you should be concerned. Those two have different answers.
I'll first answer the first to the best of my ability. I am not a lawyer, and this is not legal advice. If you are concerned, check your employment contract, and consult with a lawyer. But I can summarize the situation in two states.
In New York state, you are likely to be a professional employee. A professional employee has no set hours, and no set workplace. If you go out to dinner with a client, you're at work. If you figure out how to solve something in the shower, that belongs to your employer. As a professional employee the default, which is likely also in your contract, is that all of the software that you write is a work for hire that belongs to your employer. The prevalence of this kind of arrangement is one of the reasons that the FSF requires copyright assignment, and requires your employer to sign off on it.
In California the situation is different. As long as the software that you write does not relate to anything your employer is doing, anything done on your own time with your own equipment belongs to you, and this right cannot be signed away. However if you're independently developing the same thing that your employer is, it is probably going to be owned by your employer. Even if it was a secret project that you did not know about. Whether this case is important depends on the details of what you are working on, and what your employer is working on.
So you see that there is no simple answer. Depending on the state, and the facts of your situation, you might or might not own your own work. And if you don't own it, you have no right to license it.
Now on to the practical issue. What the law says determines what will happen should a dispute arise and be taken to a judge. However it is very uncommon in practice for disputes to arise and be taken to judges. Furthermore many employers either don't care, or positively like, their employee contributing to open source work. Particularly if the project is one that the company finds useful and they would like you to develop expertise with it. Frequently procedures exist to get OKs for you to perform such work. It doesn't hurt to check.
Furthermore if you contribute to open source work, despite the fact that it is unclear whether you have a right to do so, the odds are pretty good that you actually won't get in trouble for doing so. And if you do get into trouble, the odds are that you'll get a slap on the wrist and be told to take stuff down, rather than suffering large legal penalties. Is the remaining possible risk within your comfort zone? That is up to you. But I can tell you that there are a lot of people doing it, and fairly few stories of people running into trouble. (And in the stories that exist, usually there was some other cause of trouble, and the fallout for their open source work is a consequence.)
I am the author of the code.
This is done with a copyright notice and is completely different from the licence. Copyright says who owns it, the licence says who can use it.
Is the liability aspect completely covered?
I suggest adding something like, "In no event shall the author be liable for any claim or damages."
Does All rights reserved cover both use and distribution? If so, do most people realize that, or should I make it more clear?
It's fine, although again I suggest adding "this is proprietary software."
Ask a lawyer if it really matters to you, although in your case I think it's unnecessary.
Best Answer
Apparently, you want to retain ownership of that software.
If this is so, simply do not sign a contract that takes it away from you.
The normal case, however, is, that what you produce during your working hours is owned by your employer. Your renumeration should be compensation for that.
Maybe you can reach an agreement, where the hours you work on imporvement of that invention are not counted (and paid) as working hours. Instead, your employer agrees to buy (a license for) the improved version for approximately the same amount as you woulod make in those hours.