On DMG page 24, under "Bringing Back the Dead", it states:

A soul knows the name, alignment, and patron deity (if any) of the character attempting to revive it and might refuse to return on that basis.

• Assuming that you died without knowing the name of the cleric reviving you, will you remember their name after returning to life?


• If two strangers attempt to revive you (you reject first, accept second), will you remember both their names?

There Is No Rule

So it falls under a setting trope, but we'll come back to that.

In your first case, the character will return in the presence of the one doing the raising:

• 
  Revivify, raise dead, and resurrection require the caster to touch the body being raised.


• When cast without the dead body, true resurrection creates a new one within ten feet of the caster.

If the setting doesn't allow them to remember who brought them back, the caster in question will be close at hand to ask post-death.

In The Forgotten Realms...

A dead person becomes a petitioner. Until scooped up by their patron deity and brought to that deity's home plane, they are mindless beings. Even after reaching their patron's home plane, most petitioners remain mindless. Often, they're not even the same manner of creature as when they were alive - for example, many petitioners of The Triad eventually become lantern archons post-death.

So, in the Forgotten Realms, I'd say anything that happens while on the Fugue Plane or beyond (including attempts to resurrect) are not remembered when the recently dead ceases to be dead. Of course, how a mindless entity makes a decisions on who can raise them is a different (and unanswered) question.

In Other Settings

If you're not using a published setting, this is something you'd have to determine on your own. Unfortunately, there is no 5E source that covers it. Based on the releases to date, I don't expect there to be one. Your best bet to dig for ideas on how death works is probably 3E's Deities and Demigods.

Apparently, RAW, yes

As specified in the DMG passage you quoted, the spell bestows knowledge of the name of the caster on the target. Whether you remember it after returning to life is up to the GM because the rules don't say how much, if anything, a resurrected person remembers about being dead. By default, since it doesn't say otherwise, they'd remember everything.

The rules don't specify

That level of specificity is left up to the DM, or the table collaboratively, to discern. Which is the most fun for your table? Do that one.

Let's look at this practically:

Assuming that you died without knowing the name of the cleric reviving
you, will you remember their name after returning to life?

Given that the reviving cleric is probably present, and near to the PC when the PC returns to life, there's no reason not to learn their name if only to say thank you. Another answer has addressed parts of your question, here.