Obviously you asked about the text "The Muppet Show". You seemingly have another answer which shows thoroughly how to distort ordinary straight text lines in a resembling style.
I guess you want an exact replica with all details, but you want to start from ordinary text.
It has a red extrusion add-on, which is easily generated with Illustrator's 3D extrusion effect. The plain text, of course, can be made by modifying a writing which is made with some resembling font. The text must be converted to curves and then you must edit the letters node by node to the wanted form. Ligature SH must be made by uniting in the pathfinder S and H. The job isn't easier than redrawing it, I bet it's heavier.
I do not believe you find an easy envelope distort mesh pattern which stretches whole text lines with few clicks just to the shown forms. Editing complex envelope distortion meshes is probably even more painful than editing the letters one by one.
If for some obscure reason I must copy the logo text, I would take the Pen and draw a copy (=manual tracing). Actually I would draw it in Inkscape, because the tools there seem to fit a little better to the job. But that's only an opinion.
Illustrator's Live Trace of course can be tried, but the result will be poor when the image is a screenshot. The result will be much better if you at first clean the image in Photoshop (=remove the extras, leave only black text outlines, fill the gaps) but the needed work for perfect result will be the same as redrawing it.
The automatic tracing problem earns some attention because one can meet cases where its accuracy is enough.
So let's try it. At first the cleaning attempt. The black outline is not as bad as it generally is when somebody asks "Please, convert this to a sharp, smooth and accurate vector!" Its so solid that an image enlargener (not Photoshop's image resize!) can guess it's a sharp border and should stay thin. Enlargening to 400% with On1 Resizer and filling to a new layer with the paint bucket gives at least in small size pretty good filled letters:
The green layer was copied and pasted to both Illustrator and Inkscape. Black and white tracing (after finding a good treshold) gave in both programs quite the same result. Here's a zoomed shot from Inkscape. The fill is removed and a black stroke is inserted:
In Illustrator one can make the red extrusion. On top there's a grey version with black stroke. Here's the result:
Quite coarse, I would say! But it could be perfect if The Muppets visit as guest stars in The Flintstones.
ADD: New edits in the question show that you really want make your own text but adapt the enveloping style to some degree from The Muppets. I recommend you to sketch some cages for the letters and then stretch your letters to the cages - not exactly, but keeping a good letter spacing. You must outline the text, ungroup and use the available distortion tools such as black and white arrows and the free transform filter (expand the result). I used only the arrows in the next example:
Envelope distortion make straight lines easily curved. I wouldn't use it if my original font contains straight lines. Arrow tools retain straight lines. If you move anchors of a curve, you must also adjust the handles to keep the curvature right. That's done in my S.
I think that those decorative extras such as the hook in the M are useful. You must draw something which fits to your letters. You can join combine new parts to the old ones for ed. with the Shape Builder or Unite in the Pathfinder panel.