There is no doubt that the President wanted to clear the Mississippi, which included taking Vicksburg. What makes you say that Lincoln "wanted an attempt to take Vicksburg by storm"? How would Sherman know about such a desire even if he did have it? Sherman's knowledge is what Grant told him and what Grant ordered him to do -- not what Lincoln was talking to Halleck about.
Here, BTW, is what Sherman actually thinks he is doing on December 8, 1862:
HDQRS. RIGHT WING, ARMY OF THE TENNESSEE,
Oxford, Tenn., December 8, 1862.
Rear-Admiral DAVID D. PORTER,
Commanding U. S. Naval Forces, Cairo, Ill,:
The movement thus far has been eminently successful. General Grant's moving down directly upon the enemy's strong lines behind Tallahatchie while the Helena force appeared unexpectedly on their flank, utterly confounded them, and they are now in full retreat, and we are at a loss where they will bring up. We hope they will halt and reform behind the Yalabusha with Grenada as their center. If so, General Grant can press their front, while I am ordered to take all the spare troops from Memphis and Helena and proceed with all dispatch to Vicksburg.
Time now is the great object. We must not give time for new combinations. I know you will promptly co-operate. It will not be necessary to engage their Vicksburg batteries until I have broken all their inland communication. Then Vicksburg must be attacked by land and water. In this I will defer much to you.
My purpose will be to cut the road to Monroe, La., to Jackson, Miss., and then appear up the Yazoo, threatening the Mississippi Central road where it crosses the Big Black.
These movements will disconcert the enemy and throw them on to Meridian, especially as General Grant presses them in front. All this should be done before the winter rains make General Grant's roads impassable. I will leave for Memphis to-morrow (Tuesday) night, and will reach Memphis with one of my old divisions Friday night. We ought to leave Memphis before the 20th, and I do earnestly desire you should meet me there at all events. Even if the larger gunboats cannot proceed at once, send those of light draught down with Captains Phelps, Gwin, Shirk, or some officer to assist me in the preliminary work. Of course Vicksburg cannot be reduced till you arrive with the large gunboats.
General Grant's purpose is to take full advantage of the effects of this Tallahatchie success.
I am, with great respect,
W. T. SHERMAN,
Major-General, Commanding.
This is what Grant thinks he is doing on December 8, 1862:
HDQRS. THIRTEENTH A. C., DEPT. OF THE TENN.,
Oxford, Miss., December 8, 1862.
Brig. Gen. FRED. STEELE.
Commanding U. S. Forces, Helena, Ark.:
I have just received authority to retain all General Curtis' forces now within my department until further orders. This is from the General-in-Chief, and contemplates their being used in co-operative movements to effect the capture of Vicksburg. If these troops have gone back to Helena I wish you would return them to Friar's Point, or the most suitable place to march them directly upon Grenada, or embark them for Vicksburg, as I may decide upon. General Halleck's dispatch was only this moment received, and my mind is not fully made up as to the best method of capturing Vicksburg. I can, by the time our troops could reach there, have the railroad completed to Grenada and a supply of provisions thrown in there. From that point Jackson, Miss., could be reached without the use of the roads. Jackson once in our possession would soon insure the capitulation of Vicksburg.
I rather incline, however, to the plan of sending your forces, and all I can spare from Memphis and here, say 25,000 in addition to yours, down the Mississippi to effect a landing above Vicksburg, probably a short distance up the Yazoo, and have them co-operate with the gunboats, whilst I move south with the remainder of my forces from here.
Please inform me by return couriers all you know about the present condition of our gunboats. You having been so long on the Mississippi River looking toward Vicksburg are possessed of much information as to the best method of attacking that point that I am not possessed of. I would be very glad to have your views.
I shall send Sherman, if the Mississippi route is determined upon, and would be very glad if you could accompany him. I have no definite news of the result of Hovey's expedition. From the enemy evacuating Tallahatchie so suddenly I judged that it had proven successful. We followed up the evacuation with all dispatch, our cavalry pressing their rear all the way from the river to Coffeeville, killing and wounding many and capturing about 700. Besides this many deserted and are coming in every day.
Word was sent me to-day by a deserter who gave himself up that there were near 2,000 Southern soldiers east of this place, scattered through the country, desirous of coming in if we would send out cavalry to drive out the guerrillas that hover in that direction and make it unsafe for them to venture in this direction.
U.S. GRANT,
Major-general.
P. S.--Send me word what number of men you have river transportation for. I shall adopt the plan of sending a force down the Mississippi. I will send instructions to the commanding officer of the cavalry forces that will necessarily be left behind for their guidance
Steele's opinion on where to land:
You ask General Steele's opinion as to the best point to land troops below here with a view of attacking Vicksburg. General Steele says that the troops should be landed at the lower end of Milliken's Bend, opposite an island called My Wife's Island, say 2 or 3 miles above the Yazoo River. From that point operations could be directed as may be hereafter indicated by yourself and General Sherman. The above point is about 25 miles by water above Vicksburg and some less by land.