agree - status vs visa

Status and visa are two separate things.

I believe you entered on an H-4 visa and so you were on H-4 status. H-4 can be changed to an H-1 status and you can start working without having to leave the country. Said differently, what you have (starting from Oct. 1) is H-1 status, but NOT H-1 VISA.

VISA is for the purpose of authorizing you to enter the country only. In the future, should you travel outside the U.S., you'll need to submit the I-94 on your H1-B approval notice when you leave, and apply for an H-1 visa before you can reenter.

Hope this helps.