Refinishing teak veneer table top  

Watco Danish Oil is one of the brands that has added varnish to it.  I have used it on some stuff where I wanted more of a sheen than just teak oil gives.  I don't remember ever doing three coats in that short a time, though--is that what the directions on the can say to do?  Also, you need to go over it with steel wool between coats, especially after the first coat.  At least that's what I felt was necessary when I used it.  The wood felt a little rough after the first coat, whether it was dust caught in the finish or raised grain, I don't know.  But #0000 steel wool took care of it.
I did have to do several coats before the sheen was even.  But I also did not sand my pieces at all, they were stripped with lacquer thinner on some and alcohol on others (depending on whether they had been lacquered or shellacked.)
I suspect your problem is due to uneven sanding, though.  But maybe also not waiting long enough between coats, I'm not sure.  I am not an expert on sanding, but my guess would be that 220 grit is no fine enough to get the wood back to the super smooth surface of a patina built up over decades.  Even if the wood was stained and dinged up, it probably was worn pretty smooth.  I've noticed this effect on small areas where i had to sand some stubborn little stain out---not really noticeable enough to worry about on whatever it was I was doing, but yeah.  It can happen.
The leaves--I know people here have asked about the same thing and I've seen it on draw leaf tables myself, where the wood is lighter and brighter because it hasn't been exposed to light or use nearly as much as the main top.  Some people here have said to just leave it as is, it's just how these tables are.  I would probably not mess with it myself.  You can always get a beautiful linen tablecloth to use for those times that you need the whole table length.

I am not seeing what you are talking about in the photos, but it could be due to uneven sanding as Spanky suggested.  It could also be due to uneven stripping (perhaps a bit of old finish remained in places).  It could also be due to uneven oil application.  Did you flood the surface with Watco, or did you "stretch" the oil?  If you "stretched" it, the finish can be very uneven. 
 
At this point, I would probably try giving it a quick steel wooling per spanky's advice, then adding more oil.  Good and flooded, and then wait a bit until the oil just starts getting tacky and wipe it all off with a cotton knit cloth (like an old tee shirt).  Try another two coats or so.  
If that doesn't fix the problem it is uneven stripping or sanding.  (Oh and I can get teak quite adequately smooth with 220 grit sand paper, especially after it has been used for a minutes or so and is more like a much higher grit sand paper).

Thank you so much for reading and responding Spanky and Leif Ericson.
Leif - I did not flood. I did indeed "strech" on the Watco Teak Oil. More like wiped it on with a pad.

Question: DID YOU USE AN ORBITAL PALM SANDER!!!???
It looks to me in the one close up picure that you've created pigtails in the wood from the sander.  If that's the case you have no choice but to start over completely.  If you have created pigtails in the surface, this will make the finish absorb differently creating an uneven finish in both color and sheen.  A rule of thumb I like to stand by is always sand by hand when working with veneers and WITH THE GRAIN!  Even if you do use a palm sander, it's very important to complete the sanding process by hand to insure a smooth surface.  In the future I would avoid Citristrip, Watco Teak Oil just from personal preference / experience.
This can be fixed though...
1.) Strip the current finish.
2.) Sand starting with 120 down to 220 WITH THE GRAIN to insure a smooth surface.
3.) Apply finish in 24 hour intervals, lightly sanding with 420 in between coats.
Going forward I would look into Waterlox, Arm-R-Seal wipe on poly, and Van's beeswax furniture polish.  By far the best products I've ever used for Danish furniture without a natural tung oil finish.  3 Coats of Waterlox, 2 coast Arm-R-Seal, finish with the Van's and #0000 steel wool.
However, if you choose to continue with the Watco, please note that because it is diluted with varnish, you really should wait 24 hours between coats as spanky indicated.  Also, because you wiped it on with a pad, you added very thin layers and probably needed at least 5 coats for an even finish.

Many thanks M_Dennis87!
I think you're right I have to start the whole thing all over. I googled the pigtails you mentioned and I think this is what happened with me. But at least I will can strip both the leaves and the top in one go and aim at one uniform finish.
You said you did not recommend Citristrip. What type of a stripper would you use instead?
I actually did find some Arm-R-Seal wipe on poly at home. Just need to get Waterlox then.

One more note M_Dennis87 - I did not use orbital sander. But I am gulity of skipping grits in between and holding a handsander at an anlge and pressing down to spot sand certain areas to get some of the scratches out - hence the pigtail effect. 🙁 Utter stupidity on my part, I admit. Lesson learned.

Be very careful sanding out spots in that fashion as it is easy to go through the veneer.