Difference between ''to say something without thinking'' and ''to say something inadvertently''

The Cambridge Dictionary gives the default (most common) sense:

inadvertently adverb

• in a way that is not intentional

But Merriam-Webster adds a second sense (for the adjective; the adverbial sense follows):

inadvertent adjective ...

... 2: not focusing the mind on a matter : inattentive

The second sense here is obviously largely synonymous to 'without thinking', though both phrasings are usually used when a person says a certain thing unguardedly rather than when referring to a complete airhead.

The default sense might be considered to be synonymous here, but people usually say what they intended to say a split second before; here, it would be the ramifications that were unintended.

I'd always use 'without thinking' here, unless I let out a secret I wasn't aware was a secret.