It looks like a start to a future feature. I see nowhere that the speaker name appears. It is not exported from the transcribe tab options to export a text file. The speaker names are imported with a saved transcript.
For anyone who has not tried this: You must elect the Speakers option when transcribing. After you do, in the Transcript tab, you will see, for example, "Speaker 1" etc on the left? Left click on Speaker 1, and pick "Edit Speakers." Change each name to whatever you want for that speaker. It replaces "Speaker 1" with your edit. Same for "Speaker 2," etc.
Upvote this feature request:
https://adobe-video.uservoice.com/forums/911233-premiere-pro/suggestions/42901230-beta-add-speaker-names-and-2-lines-option-to-capti
Edit 10/4/23. This request from the User Voice system was migrated to the new Ideas forum here. See my update post here:
https://community.adobe.com/t5/premiere-pro-ideas/beta-add-speaker-names-and-2-lines-option-to-captions-from-transcription/idc-p/13998726#M10600
There are several that ask for speaker name to be included in the export to a text file. Just search "speaker" in that forum.
Stan