If you're serious about this new career, then make sure a new experience at Company B (or another company) won't end in the same result.
The reason was a mix of bad feedback from their part ...
Feedback can be really valuable. Spend some time to take it in and see if you can find instances in which your actions (or inaction) caused issues highlighted in that feedback. Can you think of ways in which you would act differently in the future to prevent the same issue from happening again?
The important part is to learn lessons from failed experiences and being able to show that you've learned and what you've learned.
When you interview again with Company B (or another company), be candid about your shortcomings while working at Company A, but at the same time talk about what you've learned and how that experience helped you become a better professional.
... and my lack of teaching related skill that I do want to improve.
That's not a deal breaker in of itself, since companies are often willing to train you and invest in you so you can become the valuable asset that they need.
Just be forthcoming about where you are right now, about what you would need or you think would help you get to the point where you should be and tell the company you're willing to learn from them if they would take you in. Motivation goes a long way, so show how passionate you are about this career.
If you're planning to teach, then perhaps more formal training would be helpful. If you're willing to take more courses (possibly on your own dime and your own time), then consider whether that could bring you better chances.
It's hard to tell whether Company B will take you. If they do, you have one shot with them, so make sure you're prepared. (You can also tell them you're aware this is your one chance.)
Whether at Company B or another company, try to elicit feedback earlier in your trial period, so you can have a better idea of where things are and adjust if necessary.