Does "my + singular noun" imply my only something/someone unless it comes with any prior reference?

Example 1: My friend told me that ……….

"My friend" is a singular noun phrase, and its plural form is "My friends", right? If so, does this sentence "My friend told me that….." imply that "I have just one friend" unless it comes with any prior reference?

Example 2: I forgot my bag in my office.

In this example, "in my office" sounds just okay to me because many office workers, generally speaking, do their work in one dedicated or main office, but how about "my bag"? I feel that the sentence implies "I have just one bag, and forgot the bag in my office." unless the "my bag" is referred to in advance.

Example 3: I have a pain in my head (or my neck, my heart, my stomach etc.)

This looks OK to me because the human being have only one neck, heart, stomach etc. Likewise, "My head (my neck, my heart etc.) aches" is also okay to me. Compare with the next example.

Example 4: I have a pain in my eye (or my ear, my hand, my leg etc.)

This looks strange to me because in general healthy people are born with two eyes/ears/hands/legs. This sentence makes sense to me only for someone who lost one of his/her eyes (or his/her ears, his/her hands, his/her legs etc.), for example. Likewise, sentences like "My eye (or my ear, my hand etc.) aches" sounds illogical to me.

Considering cases like the above Examples 3 and Examples 4, I thought "My friend" in Example 1 and "my bag" in Example 2 suggest my only friend or my only bag.

What do you think?

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POSTSCRIPT (Added on 25 July)
After my post above, I did further research on this question over the Internet. Let me add some of the information I found, for your reference.

in most dialects of English, "my" is a definite personal pronoun, so
you should use "my friend" whenever you would use "the friend", and "a
friend of mine" or "one of my friends" when you would say a friend.
https://english.stackexchange.com/questions/34164/my-friend-vs-a-friend-of-mine

"I met my friend …" means that you only have one friend, since we
don’t need any more information to identify this person.
https://forum.wordreference.com/threads/my-friend-a-friend-of-mine-formality.2238527/

"[The] construction with the possessive pronoun [e.g. a friend of mine] differs from the alternative of possessive determiner + noun
(e.g. my friend) mainly in that it is more indefinite. The sentences
in (30) below illustrates this point.

(30) a. You know John? A friend of his told me that the food served at that restaurant is awful.

(30) b. You know John? His friend told me that the food served at that restaurant is awful.

The construction with the possessive pronoun, in (30a), can be used if the speaker hasn’t specified and doesn’t need to specify the
identity of the friend. In contrast, the construction with the
possessive determiner, in (30b), implies that the speaker and listener
both know what friend is intended."
https://www.thoughtco.com/possessive-pronoun-1691649