Does the Statement name change when going from pending to cleared?

Yep, if your bank downloads the pending transaction and then downloads the cleared transaction a day or so later, it overwrites the pending. My bank also changes the date of the transaction.

Mine changes a lot and it adds a transaction number.

Well, every bank is slightly different. I only have 3 banks that download pendings. If you have a rule for the pending to change payee or mark as recurring, the second download should recognize it (being the same amount). I usually don't have problems with the cleared transactions matching the already downloaded pending but I do sometimes end up with a duplicate and I merge if I can.

What happens with my local bank is this:

Pending downloads on Day 1

Pending redownloads on Day 2, sometimes matches, sometimes not. I delete the second one (duplicate)

Cleared transaction downloads on Day 3 and matches with the Pending.

My 2 credit cards (Amazon Visa and AMEX) only download the pending once and then the cleared. They usually match no problem. So have never paid much attention to the statement name. I will be on the lookout to see if they change description name as well.