SATA SSD is the size of your external hard disk @ 2.5".
M.2 = the long stick kind. The one which is in your MBA now. M.2 kinda means that your SSD's width is 22cm.

You will need either M.2 NVMe vs M.2 SATA.
SATA = generally your HDD's speed maxed at 600MB/s.

That doesn't mean that if you buy M.2 SATA SSD that the speed is going to be similar to a HDD. I am not sure how but of course M.2 SATA SSD is going to be way faster than a HDD.

SATA and PCIe NVMe are bus types.
That's why you will see PCIe NVMe M.2 SSD or NVMe M.2 SSD or rarely PCIe M.2 SSD.
It's like telling everyone that you are using the best highway (PCIe NVMe), driving the best car design (M.2).

SATA M.2 supports Key M and Key B.
NVMe M.2 supports key M only.

You can install SATA M.2 Key M + B on NVMe slot. NVMe SSD CANNOT install into SATA M.2 slot. Not a chance.

Sorry back to your case.
Need to have adapter -> YES if you are going with like mainstream brands. They are not Key M nor Key B.
Which type of SSD to choose -> Just go for NVMe M.2 SSDs if you are going with an adaptor.

Different types of SSD Explained
EveryMac MBA SSD Upgrade Guide
EveryMac your MBA specs. Head to Standard Storage to view more.

I don't own a mac but I know a little thing or two about it. I can't answer on the performance but definitely without adapter is 100% going to be faster. Hope something helps here!

I bought this last time for my friend's MBA storage upgrade.
Everything is plug n play.
But you need to prepare the MacOS installer file in a thumb drive.

Transcend SSD
https://shopee.com.my/Transcend-MTE220S-NVM...9824.6713616509

SSD Adaptor
https://shopee.com.my/ACASIS-Apple-MacBook-...0473.9910617919