Great question, Sarah!
I’ve come across both of those pronunciations, and some more. Here are three different examples of the varying pronunciations of the same name:
Since you mentioned an Esoteric Mage, I’m assuming you’re working from a Western Esoteric perspective. Here’s a video on that topic. The pronunciation of Thoth they use is similar to “both” but pronouncing both “th’s”.
Israel Regardie: The Golden Dawn Tapes Vol I: Lectures on Egyptian Gods - YouTube (pronounced at 12:28)
According to Wikipedia, the two most common pronunciations are: /θoʊθ/ and /toʊt/; You can read them with an IPA reader such as this one: http://ipa-reader.xyz/
I believe that we should try to learn the proper pronunciations as much as possible, as a sign of respect. However words tend to change, and this word is thousands of years old. It went from Ancient Egyptian through Coptic, through Greek until it was Latinized. And here we are trying to say it in English.
My conclusion is that a tomato is a tomato no matter how you pronounce it. Pick the pronunciation that is easiest for you and practice it so that you can say it confidently in ritual or in conversation. If you are working with Thoth (such as in the Thoth wisdom devotional), you may adapt the pronunciation or meditate on the name to see if you receive/hear a different pronunciation in your personal work.