Based on your description, you have not been terminated. Basically, they have decided not to renew you after the probationary period. However, it will depend on what is said on the initial documents you received when you joined. Although this doesn't make an apparent difference to you in terms of your current employment status, what is important to note is that they have simply decided not to keep you but not 'fired' you. It could be due to various reasons, which may or may not be because of your performance.
I would suggest that you have a conversation with your (ex) manager and try and get more information. If he gives you a good reason for not renewing your contract (something that is not related to you - for example downsizing) and is happy about your performance, it won't hurt to use him as a referee for a new job. Many potential employers are keener to know about your more recent work experience.
And with regard to receiving something in writing, it will depend on what the original contract says. If the contract says it ends by a particular date, then to me it sounds like they are not required to provide you with a letter of termination, because as I said before, technically it does not sound to me like a termination. Also, have a look at the relevant laws applicable to your jurisdiction if there's any ambiguity.
On a personal note, try and stay positive, get constructive feedback from your manager and good luck on your new journey!