I've recently decided to re-teach myself how to play the piano, but I'm not sure what type of keyboard/digital piano I should be looking into purchasing.
I'm also on a budget which is why I'm leaning more towards a keyboard rather than a digital piano.
You re-teach, it means you used to play ? What is your experience and level. On what instrument you made your first lessons ? Enough to have some abit/taste in term of action feeling ?
Then, as KJ asked, what is your ultimate goal ? What is appealing / fun for you re-teaching playing piano ?
I mean, you dream of being like a classical solist on his concert grand, giving recital of bach, mozart, chopin. Jut for yourself ? For your family ? So action realism, purity of sound, speakers power, cabinetry are important.
Do you want to have fun playing pop music you like, perhaps as singer accompaniement ? Perhaps, you want to play for other, or be a perfromer in a band ? Compose your own music ? So portability, connectivity and stusio recording, rythm box, versatility of sounds, is important.
If you have enough budget, you should try to buy something aligned with your "dream". And try to touch / approach your dream from the start.
I think, restart is not so easy, so you need to have fun, and do something you like, when you will start to spend a lot of time to "work". You need to keep motivated.
I do not advice you to buy the "perfect" / most expensive DP, but just something (possibly reasonnably priced), that is "serious" on your way to fulffill your final motivation, and that you enjoy when your practise.
For example, if you need to connect things, prepare this and this, this is just little pains, but it can prevent you to spend 10 minutes of playing. If just because you walk near you instrument, and just let your fingers runs on keys, and finally, you sit down and continue, because you had immediat access to instrument, and it was just cool enough to prevent your escape, it is a godd spec.
I fear starting with "fake" / toy / unconfortable instrument, is a way to concede it is not important for you, and you can lose time, or just make a trial. So for me, a bad start.
That is generally about being serious, having faith in your re-start projetc. Then, if I can express a personnal opinion, the most important, when you learn, is to have a good action to teach/train your fingers to have strenght, sensitivity, velocity of a "real/standard" keybed, so you can adapt/evole to the better instruments, trying to reach better expressiveness (and this is independant of what kind of music you will play). So, if you are "budget-limited", if you can save some money with the cabinetry (and be confortable having a slab), then it is worth to put savings for a better keybed, one that suits you and make you connected with the instrument (if you have sufficent experienced to have a taste of that). If you don't have preferences, try to follow general consensus about good action for budget DP (you have several options, depending of personal preferences, for me Casio tri sensor II, or yamaha GH, or Roland PHA-IV, or Kawai RH2). Casio privia PX-150 being the cheapest in the "quite good" action league.
Just my 2 cents, after seing my young daughter (8) started the piano, and see how I (40) re-started (or not) myself as an adult with piano (or continue practising guitar), when time is rare and distractions countless.
At the end of the day, you really need, is to "like" your instrument, enjoy using it and be certain that the quality is good enough to grow up, so it is not an "barrier" to your music, it is a reliable friend, not an ennemy (actually, the only remaining problem should be you and your work
Hope it helps and not confuse even more ...