I’ve been out of the hobby for a long time, so I’m probably a bit out of date. But it is worth summarising some of the history.
Dobsonian should refer to the design developed and popularised by John Dobson. He had a specific intent - make viewing of the astronomical sky accessible to everyone. In an age when “proper” telescopes emphasised precision parts, equatorial mounts, obsessive accuracy of optical components, he went the other way. Make it cheap, easy to make in anyone’s garage, and most of all, deliver good views to the naked eyeball.
His design was nothing more than a wooden box with a relativity thin, but big, mirror mounted via simple sliding bearing into another box that itself rotated on Teflon pads on a sheet of wood. But he developed it over the years and to the surprise of the established amateur telescope making community these cheap light buckets delivered on the promise.
Now the basic design is a mainstay of available telescopes, and sits alongside other more traditional telescope designs. It has also formed the basis of much more advanced and feature filled (and vastly more expensive) offerings.
It is possible to make a tracking Dobsonian mount, and I remember in the early days when the amateur telescope making community began to embrace the idea there were a number of clever designs that provided anything up to an hour of basic equatorial movement with careful layout of bearing surfaces and little more than an electric clock motor and threaded rod. The success of these designs depends a bit on your latitude. Computer controlled alt-az is possible, but there is a point where you cease to be meeting John’s original objective of cheap accessible sky viewing. IMHO a true Dobsonian should deliver a cheap fun light bucket. Additional funds should go into more aperture.
Equatorial mounts were initially the thing because you can track the sky with only movement in one axis. If you have computer controlled motor drive on both axes you can of course track the sky with an alt-az mount. The only downside is that the field of view rotates over time, so photography becomes impossible unless you rotate the camera, and that means another computer controlled motor. But computer control is now near free. No modern scientific telescope built will be on an equatorial mount. The sheer size of modern devices would make that impossible anyway. A less commonly mentioned issue with mounts is the amount of sky they afford easy view of. This depends upon your latitude, the size of the telescope tube, and overall balance of the design. Alt-az can win big here.
Hobby level telescope are a mix. You see alt-az mount scopes, equatorial mounts, both with and without motor drive and computer control. You can argue the value proposition forever.
Worth mentioning, if you use a basic telescope, you can use a smartphone sky view app as a pushto guider. For Android the SkEye app is designed for purpose. Strap your phone to the scope and it will tell you how to point it at whatever you need, at least within range of being able to visually pick it up. That plus a good sized simple Dobsonian might bring about the most joy for the dollar.