There really is no one right answer here. I learned to play keyboard on a home organ, which had two manuals (keyboards) of light-touch, synth-style keys as well as a third smaller monophonic (can only play 1 note at a time) synthesizer-type keyboard. I also learned bass pedals on the organ. So, no, I did not learn on an 88-key weighted-key instrument, initially. But what I learned on organ helped and mostly carried over when I did start playing around with real acoustic pianos.
Some of the big differences had less to do with the number and weight of the keys, and more to do with the way the instrument works. On the organ, I could just hold a chord with my left hand for as long as I wanted while playing a melody with my right hand, and then using the bass pedals for the, well, bass. On a piano, if you just hold down a chord, the sound will die out in a few seconds, so you have to keep playing something with your left hand to keep the accompaniment sound going. And of course, without any bass pedals, you have to also be mindful of playing bass notes with your left hand when playing a regular piano.
As for weighted keys versus synthesizer-style, light-touch keys, one is not better than another -- they just have different purposes. Yes, if you ultimately want to be able to play an acoustic piano, then weighted keys are good to get you used to that key feel. And there is certainly a certain amount of expressiveness that you get with weighted keys. On the other hand, with synth-type keys, with a well-designed keyboard, as you get good at playing, you can really whizz through fast, complex melodies and passages.
I strongly disagree with the need to spend $1000 or more on a keyboard to get a good sound, but again, this depends on your goal. Yes, you will get a better built in sequencer/recorder, more sounds, some higher-quality sounds, more and higher-quality styles, and other features like multi-pads. But I can tell you that I have been playing keyboard 41 years (though not professionally), and I have owned keyboards like the home organ I mentioned before as well as a Yamaha DX-7 synthesizer, and I had experience with other pro synthesizers (Arp Odyssey and Arp Omni 2) when I took an electronic music class in high school, and yet my main keyboard right now is a Yamaha PSR-E433 that I paid $250 for. I do multi-track recordings using its built-in rudimentary 6-track sequencer, but then I send that to a computer as audio using Audacity, and then I can add additional audio tracks directly on Audacity.
And, of course, as has been said or implied, another nice thing about learning and playing on an arranger keyboard, versus a standard piano, is the wide number of sounds and styles available to you on the arranger. Just be careful not to get too distracted by these features as you're learning, because as you first start out, you want to concentrate more on learning scales, chords, reading music, music theory, and simply how to consistently hit the correct notes when you're playing a song, and not get too distracted by playing around with the buttons. You can experiment with different sounds, of course -- just don't get too engrossed in that aspect of the keyboard too early if your goal is to actually learn how to play keyboard or piano.
A good set of keyboards that, as I understand it (and someone correct me if I'm wrong), combines many of the features of arranger keyboards (dozens of sounds and styles, built-in recording) and has 88 weighted keys, would be the DGX line of Yamaha keyboards, like the DGX-660, and these are usually well under $1000.