Sifting Through Processor B.S.

an P-M 1.7 is as fast as P4 @2.8

it is a speedy processior

Best processor is a bit relative but all in all, there are two processors I would consider buying - if you want very good power capabilities and long battery life you DO NOT have an alternative to the Dothan (new P-M). If battery life is not such a concern to you and RAW power is what you need go for the AMD 64 - this is the next most efficient processor after the Dothan.
P4 is a powerful cpu, but not as much as AMD64 - except in multitasking where the P4 HT is SLIGHTLY(hint) ahead - yet the P4 HT is the least efficient cpu out there and battery life is about 1-1.5hrs with it. Furthermore, P4 based laptops are the heaviest and least portable, while the lightest are Dothan based laptops. AMD is in the middle.
Basically, get:
Dothan - if you want long battery life, light notebook, very good power capabilities in your cpu (a Dothan 2.0 performs more or less like a 3.0-3.2 P4HT, except in multitasking), cool running temps.
AMD 64- if you want RAW power and decent batery life combined with not that high running temps in a not that heavy laptop
P4 HT - if you want RAW power with limited battery life (1-1.5 hrs, at best) and it can get very, very HOT when under load, all this combined in the heaviest laptops out there - it is not that you can't lift them but if you are going to carry it around for several hours you may be in trouble.

AMD64 is faster than the Dothan. But keep in mind, that doesn't mean the dothan is slow by any means. For most people Dothan would be best...unless you need all the power you can get with mobility, then AMD64 makes sense. Performance vs heat/battery life Dothan wins hands down in my opinion, but then again it is designed just as mobile processor, so that makes sense.

I inserted numbers above to clarify your assumptions, here they correspond with above (duh):

1) P-M isn't really "based" on the PIII. It is much more powerful, has way more Cache (esp in the Dothan) and is all in all much better than the PIII. It is definetly not a "step up from Celeron." It is in it's own league. It is probably the most "viable" notebook solution - in that it is the only true mobile processor out there. You can expect 4-5hrs battery life on most systems (varies w/ usage/specs/battery size).

2) P4 is definetly a performer - however for many (especially on this forum), the loses to battery life and the significant heat issues are too much to sacrifice for the marginal "oomph" over the P-M Dothan. Expect 1-1.5 hrs battery life.

3) HT is designed for mulit-tasking, but that is not to say that it is vastly superior to other processors in that aspect of computing. It will definetly show in benchmarks, but most likely be far less noticeable in real world situations. Especially if you have a high Ghz P-M Dothan.

4 and 5) AMD64s have been showing the best benchmark numbers. They also provide better battery life than P4s (especially in Low Voltage versions). They are "future-proof" in that they are already in the 64bit world (expect 64bit windows OS and apps in 1-2 years). The difference in multi-tasking between in a good AMD64 and a comparable P4-HT is to be debated, although IMO, probably negligible. Expect 2-3hrs battery life, maybe more on LVs.

And one last point of my own: As many have and probably will mention, Don't get too caught up in processor type/speed. None of these processors will be a "dog" or "slow down" your system. If you want mobility, go with Dothan or AMD64. If you are going to be plugged and don't mind a (probably) hot clunker, go with P4-HT. But in the end make sure you get enough RAM, HD Space AND speed, GPU, and other components to meet your needs.

Also note that if you have an AMD64 chip and install XP service pack 2 you will be better protected against nasty viruses and worms thanks to AMD's NX bit protection. Intel has yet to implement this.

If you buy an Athlon 64 you not only get a better processor than the p4 in terms of performance battery life and (now), virus protection, you also get it at a fraction of the cost.

AMD chips are made well and offered at a fair price.

The P6 core (Pentium Pro/P3/Banias/Dothan) may be long in the tooth, but it's always been designed very well. And it's hardly a Pentium III, as a Dothan at 1.7GHz pretty much smacks around a P4 running at 2.6GHz.

I'd say an AMD64 is the best choice for you.