Fwiw, the Wikipedia article divides the meaning into three categories that more or less correspond to the three senses I proposed in comment #10.
That article has a heading for each category/sense:
1. 旅行先の土産
2. 手土産
3. 比喩

The third category mentions two subsenses. Both of them have negative connotations, as is the case with daijs sense 3 and my GG5 example.
I'll quote the 比喩 section of the Wikipedia article in full:
進物全般について土産と称することから、以下の事象についてもこの呼称が用いられる。
- 行為の本来の目的ではない物事を受け取ること。受け取る側が望んでいない物事に対して言う場合が多い。
- いわゆるリベート、賄賂のこと。

The current JMDict definition, on the other hand, suggests that a 土産 is always something nice: either a gift/present/souvenir or, figuratively, a piece of good news.

ok, although it doesn't seem to cover the "daijs sense 3" very well
(which is also the sense in the GG5 example I quoted)

I feel that Richard's proposed extra senses are related. I'm suggesting a merge of them.
I'm approving this, and reopening for any further discussion.

@@ -20,0 +21,4 @@
+<sense>
+<pos>&n;</pos>
+<gloss>present or good news brought by a visitor</gloss>
+</sense>

1) The Present Suggestion
I was thinking of something slightly longer (as is my wont, I'm afraid):
"local product bought while on a trip and brought back to give to others". 
But I think Jim's latest suggestion is pretty good. I'd be satisfied with that. 

2) Additional Senses
I think Rene is right to say that more senses may need to be added. But I also feel that, while a short definition like "present; souvenir" covers all bases, it does so at the cost of giving a misleading impression.

Regarding koj sense 2, I have already suggested (in submission #3 below) adding a gloss like "present brought by a visitor".

3) (shinmeikai sense 2) & (daijs sense 3)
GG5 gives this example:
とんだおみやげを持たされた. A hell of a present has been dumped in my lap.

In other dictionary examples, a miyage is
- a piece of good news to pass on when one visits one's parents
- a graduation diploma one has earned, that one can show one's folks back in one's hometown

Maybe something like the following gloss could be used for cases like those:
"something perceived ironically or figuratively as a gift or souvenir"

4) So I'm thinking of 
[1] local product bought while on a trip and brought back to give to others 
[2] present brought by a visitor
[3] something perceived ironically or figuratively as a gift or souvenir

Feel free to take whatever you want from that, or ignore it :-)

koj sense 2:
人の家を訪問する時に持って行く贈物。てみやげ。
shinmeikai sense 2:
何かをした際に得た良い結果で、その人や周囲にとって さらに輝かしい将来を約束するもの。
daijs sense 3:
迷惑なもらい物を冗談めかしていう語。「伝染病という、とんだ土産をもって帰国した」

i'm fine with the revised gloss, but:
*also means 手土産, which we gloss as "present (brought by a visitor)"
*also means "souvenir" in a figurative sense
*also means "souvenir" in an ironic sense

so the existing glosses of "souvenir;present" seemed to cover all bases.  if the gloss becomes more specific, then more senses need to be added

Marcus (in a forked thread): how's "souvenir (as a present)"?
Does this bring it together?

@@ -19,3 +19 @@
-<gloss>local specialty bought as a gift</gloss>
-<gloss>present</gloss>
-<gloss>souvenir</gloss>
+<gloss>local specialty or souvenir bought as a gift while travelling</gloss>

@Rene: See point (a) in my original submission, for example. 
@Jim: I think that's not bad, although it omits aspects of the meaning mentioned in Japanese definitions. E.g. "旅行先などで".
(スーパー大辞林, Wikipedia, shinmeikai etc.)
I suppose you could say "souvenir bought as a gift" (instead of just "souvenir"). That would eliminate souvenirs that one keeps for oneself, as in "I bought this little pot as a souvenir [memento] of Spain."

Does this work? It goes beyond present/souvenir.

@@ -19 +19 @@
-<gloss>local specialty (packaged food, or handicraft etc.) bought while one is on a trip, and given, on one's return, as a gift to a friend, coworker or relative</gloss>
+<gloss>local specialty bought as a gift</gloss>

i mean, the thing is that i think that description is pretty much what a souvenir is.  when you buy someone a souvenir, you're not usually buying them the kind of thing they can get back home or from somewhere else.  that's the whole point of buying it while you're away.  if you're in paris and you buy an eiffel tower keychain, you call it a souvenir; if you buy a fuzzy stuffed flamingo, it's just a gift

I've been trying to make it shorter. Here's one suggestion: 
"gift, bought while on a trip, that is representative of the place to which one traveled"

The Wikipedia article says 土産（みやげ、みあげ、どさん、とさん）は、知人や縁者に配る目的で旅行先などで買い求めるその土地にちなむ品物（進物）のこと。または知人や縁者の家宅など訪問先を訪問する際に感謝を込めて持参する進物のこと。後者の場合は手土産（てみや
げ）という言い方もする。 So they distinguish two senses: 
1. something like the one I suggested in my submission 
2. another one that could be defined the same way EDICT defines 手土産: "a present brought by a visitor"
https://ja.wikipedia.org/wiki/土産

Of course, the definition I submitted is wordy. I can understand you might want to either trim it down or reject it entirely. But I feel that neither of the senses mentioned in the Wikipedia article is adequately conveyed by the current 
JMdict definition of 土産 ("present; souvenir"). If JMdict editors disagree with me, though, I will accept it.

shinmeikai
外出先から家へ帰る時や人の家を訪問する時などに持っていくちょっとした品物。

this seems needlessly specific, descriptive and encyclopedic to me

This submission was prompted by working on a definition for a Chinese word, used primarily in Taiwan, which is very close in meaning to Japanese "omiyage", namely 伴手禮. I consulted Japanese dictionaries looking 
for inspiration.

Neither "present" nor "souvenir" seem very apt for 土産. An omiyage is something that combines elements of both "present" and "souvenir" but doesn't mean either of those things.

> souvenir : "something kept as a reminder (as of a place one has visited)" [Merriam-Webster]
(Unlike an omiyage, it's usually something one keeps for oneself.)

> present : "thing given to someone as a gift" [Oxford]
(Unlike an omiyage, it's not typically associated with a trip to a particular place.)

Some quotes
a) The ルミナス和英辞典 entry for 土産 says 【日英比較】 日本語の「土産」にぴったりの語はない. souvenir は必ずしも他人への贈り物とは限らない.
b) "Ever heard of omiyage? Well, when a Japanese person tells you that it means “souvenir,” you’ll probably be picturing a keychain or a shot glass, right? Oh, how this translation misguides you."
https://www.fluentu.com/blog/japanese/omiyage/
c) "Learning Japanese in university, one of the first words I was taught was omiyage (お土産). Back then I was told that omiyage meant ‘souvenir’ but let me assure you– omiyage is not a mere Japanese souvenir."
http://www.travelsintranslation.com/2014/03/omiyage-culture-japan/
d) "A souvenir is usually something you buy for yourself to remember the trip. ... A Japanese omiyage is different from a souvenir. It’s not something you buy for yourself, but solely for others."
https://zoomingjapan.com/culture/omiyage-culture-in-japan/

スーパー大辞林 3.0 says, in part:
旅行先や外出先から家などへ持って帰るその土地の産物。

@@ -18,0 +19 @@
+<gloss>local specialty (packaged food, or handicraft etc.) bought while one is on a trip, and given, on one's return, as a gift to a friend, coworker or relative</gloss>