Does the word "apparently" imply that I personally do or don't believe the statement following it?

Use of the predicate appear, or its adverb form apparently, has a Gricean interpretation.

Because of Grice's Quantity Maxim for conversational discourse

Make your contribution as informative as is required.

if, instead of simply saying

• XYZ.

one says any of

• It seems like XYZ.
• It appears to be XYZ.
• It's apparently XYZ.
• It's said to be XYZ.
• I think it's XYZ.

the implication is that that's all one knows, and one takes no responsibility for the actual truth of XYZ.

However, the reason for this failure to validate XYZ is not given; it could be for either of the two reasons given in the question, or others unknown.

In the US, most listeners would probably take away the first interpretation.

The term apparently is a hedge to indicate that the speaker is not certain or does not have irrefutable evidence of the position, but either believes it is so or is willing to accept it for the sake of argument. It is akin to saying

It [now] appears to me that . . .

If one did not believe something was so, but wished to consider or discuss it, she or he might say Allegedly, xyz . . which implies that someone else has averred it, but the speaker is, or may be, doubtful. While reporters (and lawyers) use this locution to appear neutral, most others using it would likely be indicating skepticism. It is more like saying

Someone else is claiming that . . . [and I am doubtful, or at best neutral]