Macbook pro 13 inch, Retina, core i5.. 2.6 late 2013.. A1502-2678

There are only three things it can be:

• SSD Adapter (bad or incompatible)
• SSD (bad or incompatible)
• Logic Board (bad)

Most likely it is a bad or incompatible SSD Adapter. From reading these forums for many years, the most compatible & reliable SSD adapter appears to be from Sintech. Just because someone says something is compatible does not make it so. Besides, someone making an SSD adapter for an Apple proprietary SSD connector has no other place to use it other than a Mac. The majority of items for sale on Amazon (or any other web marketplace) are absolute junk or fakes. You must be extremely careful. Even the customer reviews cannot be trusted, since so many people have been paid to make positive reviews for various products.

Keep in mind you are using an SSD that is not made specifically for this Mac. You will not get any support from the SSD manufacturer if the SSD has problems while installed internally in a Mac. Unlike hard drives, SSDs are not always compatible with all computers. People do use standard M.2 SSDs internally installed into Retina laptops, but you do have to accept the risk of issues using unsupported configurations because those SSD manufacturers will not provide support for a Mac, nor will Apple provide support for you when using a third party SSD & adapter. The only SSDs with full support from the manufacturer are from OWC since they have been made specifically for the Mac and does not require an adapter.

What SSD adapter are you using? From reading these forums for the last few years, it seems only the Sintech adapter is recommend as it is the most compatible and reliable. Or maybe the new SSD just is not compatible with this laptop. Not all SSDs will work in all computers.

Since you were able to install macOS Big Sur to the SSD while the NVMe SSD was connected externally, this should mean the laptop's system firmware has had its firmware updated to a version compatible with an NVME SSD.

Only OWC provides an SSD as a drop in replacement for the original Apple SSD and does not require any adapters.

If you are going to use an M.2 SSD, then you will need to use an adapter to convert the Apple proprietary SSD connector to M.2. I've seen people mention using a lot of different M.2 SSD brands, but I don't know which ones are best for an Apple Retina laptop. I think the SSD adapter is the critical part in most cases to ensure compatibility. You already have an SSD, so try the Sintech adapter as that is cheaper than the SSD so it makes sense to try the Sintech adapter first.

I'm not familiar with TimeTec. I'm very hesitant to use off brand SSDs as they usually take a lot of shortcuts and tend to be lower end models with poor performance. Definitely surprised anyone else making SSD with the Apple proprietary SSD adapter as there is not really much of a market for them.

Looking up TimeTec, I only found one reference which mentions it is a LiteOn OEM which is part of Kioxia (formerly known as Toshiba Memory), so it looks like this may be a legit and decent product.

Glad you got it working.