_{People seem to dislike this answer, so let me emphasise the part that is the factual answer, before we get to the proposed solution…}

You can't disable System Notifications.

^{Right. Facts done with, back to the suggested solution…}

So, instead, just do the updates. Your system is so far out of support that they will likely be the last updates you ever see.

The control panels in your top photo didn't appear until Mojave [the same time as system updates moved from the App Store to a dedicated Updates control panel], so you don't have the option of switching it off.

It goes without saying that a computer without current security updates is not safe on the internet.

_{People seem to dislike this answer, so let me emphasise the part that is the factual answer, before we get to the proposed solution…}

You can't disable System Notifications.

^{Right. Facts done with, back to the suggested solution…}

So, instead, just do the updates. Your system is so far out of support that they will likely be the last updates you ever see.

The control panels in your top photo didn't appear until Mojave [the same time as system updates moved from the App Store to a dedicated Updates control panel], so you don't have the option of switching it off.

It goes without saying that a computer without current security updates is not safe on the internet.

You can right click & Hide Update, but I don't know if that will work for System Updates.

_{People seem to dislike this answer, so let me emphasise the part that is the factual answer, before we get to the proposed solution…}

You can't disable System Notifications.

^{Right. Facts done with, back to the suggested solution…}

So, instead, just do the updates. Your system is so far out of support that they will likely be the last updates you ever see.

The options in your top photo didn't appear until Mojave.

It goes without saying that a computer without current security updates is not safe on the internet.

_{People seem to dislike this answer, so let me emphasise the part that is the factual answer, before we get to the proposed solution…}

You can't disable System Notifications.

^{Right. Facts done with, back to the suggested solution…}

So, instead, just do the updates. Your system is so far out of support that they will likely be the last updates you ever see.

The control panels in your top photo didn't appear until Mojave [the same time as system updates moved from the App Store to a dedicated Updates control panel], so you don't have the option of switching it off.

It goes without saying that a computer without current security updates is not safe on the internet.

You can't disable System Notifications.

So, instead, just do the updates. Your system is so far out of support that they will likely be the last updates you ever see.

The options in your top photo didn't appear until Mojave.

It goes without saying that a computer without current security updates is not safe on the internet.

_{People seem to dislike this answer, so let me emphasise the part that is the factual answer, before we get to the proposed solution…}

You can't disable System Notifications.

^{Right. Facts done with, back to the suggested solution…}

So, instead, just do the updates. Your system is so far out of support that they will likely be the last updates you ever see.

The options in your top photo didn't appear until Mojave.

It goes without saying that a computer without current security updates is not safe on the internet.