Photo migration

You wrote: "I fear that in migrating the Photo App in the Mac, I will erase or overwrite the existing photos that I now have on my MacBook Pro, .. The question is how to insure that in migrating the Mac Photo App with all it's photos sorted in all the albums, I am not causing a loss of any photos in my MacBook Pro. "

Where are the albums on the Mac that you want to migrate to the MacBook Pro?

Are this albums you created in Photos for Mac or folders in the Finder? Albums created by Photos will be stored by Photos in the Photos Library on the Mac, nicely bundled as a package, the Photos Library. You can transfer the Photos Library as one entity to a different Mac. By default, the Photos Library will be located in your Pictures folder. Look for it in the Pictures folder.

To use this Photos Library on your MacBook Pro you have to copy it to the MacBook Pro, without overwriting the current Photos Library there. Yer_Man told you how to do it, copy it to a subfolder of Pictures, if it has the same name as the existing Photos Library on the MacBook Pro.

There are two things to consider, when working with Photos Libraries:

• The system versions of the Macs have to match. Are the system versions on the Mac where you are copying from and the MacBook Pro the same? The Photos.app is a part of the system, and it cannot open Photos Libraries that have been opened by a newer version of Photos. So your MacBook Pro will only be able to work with the copied library, if the system version on the MacBook Pro is the same as on the other Mac or is newer, but not, if it is older.
• Copying a Photos Library to a different Mac does not merge the libraries automatically into one. You can switch between the two libraries and view them in turn, or import photos from one library into the other.

You can have several Photos libraries and use them in turn, as described in the user guide: Create additional photo libraries in Photos on Mac – Apple Support (UK)

If your Mac is running macOS 12 Monterey, you can import photos from one library directly into your existing Photos Library, as described here, to merge the two libraries: Import photos from another library in Photos on Mac

If you move the library to the same location as the existing library then yes, but not before you're asked to confirm.

Simple solution: rename the library you're moving or put in a folder within the Pictures Folder.

What folder are you talking about?

A Photos installation is made up of two main parts: the Application, part of the OS. And the Library, where your images are stored. If you move the library then you move the photos and the organisation.

The one exception is if you are running a referenced library.... are you?

A Managed Library, is the default setting, and Photos copies files into the Photos  Library when Importing. The files are then stored within the Library package

A Referenced Library is when Photos is NOT copying the files into the Photos Library when importing because you made a change at Photo -> Preferences -> General. (You unchecked the option to copy files into the Library on import) The files are then stored where ever you put them and not in the Library package. In this scenario you are responsible for the File Management.

I want to move all the photos in their specific albums from a Mac to my MacBook Pro.

Then you want to move the Library

The question is how to insure that in migrating the Mac Photo App with all it's photos sorted in all the albums, I am not causing a loss of any photos in my MacBook Pro.

Rename the Library or put it in a different folder.

You can't move the application. Only the library of photographs.

Why do you want to move the app? It's part of the OS and it should be on the other machine.