In every business situation -- which an employee - employer one certainly is -- trust is your #1 asset. You build it over time, with consistency, by always doing a great job. After some time, people will start to notice, and begin to trust you.
When I was freelancing, it was a common practice of serious clients to start a new project with an unrelated test or a small milestone. It was to make sure we can work together, and this practice allowed the building of trust without taking much risk.
I was forced to do great work every time, no matter how small the project was, as I did not know if that was a test before a bigger gig or just a cheap client.
You earn trust by keeping your word. If you promise something and don't deliver, people will not trust you.
Promise a little less, and deliver a little more. For example, you can do that with the schedule. Add a little margin to your estimates, making it more likely you can stick to it. As estimates usually fall short, being a little pessimistic helps with accuracy too.
Marketers usually do the opposite: they overpromise, but usually underdeliver. It's a great strategy if you want a sale. Pump up the client's enthusiasm, and they are more likely to choose you. But this practice does not build trust in the long run.
As a developer, you should aim for the trust, and don't settle for short-term gains. Play the long game, and play it right.
Trust needs time to build up, but only a few blows to tear down. Therefore, consistency is the key. It's not enough to deliver great work once; you need to do it over and over again.
Consistency is the result of habits, in this case, work habits. To consistently do great work, you need to develop the habits that support this goal.
Testing your code before shipping it is one such habit you should definitely embrace. If the client tries out your work and sees that it works and you also handled edge cases too, you will be seen as a reliable developer. On the other hand, if multiple QA rounds are needed every time you say you are ready, your schedules will be regarded as unreliable.
Another habit is to communicate often. Especially in a remote setting, regular communication is essential for building trust.
Last but not least, help your fellow developers by producing readable code and documentation. If your work is easy to read and you adhere to the coding guidelines and best practices, people will more likely want to work with you. This is the trust that you are looking for from a developer.
If you embrace trust-building habits, people will notice it eventually. Treat it like a bank account: you need to accumulate enough social capital in order to reap its benefits.
Trust is your #1 asset, and you should prioritize it. Think about it: even great experts in your field is in a worse position if they've never provided to the company. Your employer gives you salary because they trust you to do your job well. You earned their trust.
But in order to earn it, you need to do great work consistently. Build working habits that ensure you are doing a great job.
Trust is earned the hard way; there are no shortcuts to it. Work for it.