I do not differentiate between training shoes and competition shoes, I train in and compete using the same shoes.
Weightlifting Shoes – I have long been an Adidas aficionado, ever since the Cadet sneaker was released in the USA in 1974. Lifting shoes from them were becoming the international standard among elite lifters at the time but were unavailable in the USA. When my 16 year old self called the Adidas offices in Germany on a landline to try to order a pair, it was a half day ordeal that failed. I got my first pair, used, via a trade, actually two trades. I traded Stu Thornberg for a pair of Adidas lifting shoes that he had traded from Daniel Zayas for some US clothing. I had and used those shoes for over three decades. In the end I just brought them out for special occasions. At the 1999 World Championships (I was there attending the International Weightlifting Conference), Adidas were selling their newest lifting shoe for an insanely low price. Understanding that my current pair was on its last legs I picked up a new pair. Those lasted about a decade and a half, with minor repairs periodically until the synthetic part of the toe box cracked completely through. My current pair is a 2010s Adidas Powerlift shoe, their cheapest model ($89 at the time). After about 3 years of use I started having to re-glue parts of the bottom layer of the sole back in place, as there were random separations from the next layer up. A couple years ago I removed the entire bottom layer (the crepe rubber layer), sanded the two facing surfaces clean, Shoe-Gooed them uniformly, and then clamped everything together. They are back to working nicely. Adidas offers several models of lifting shoes at present, but I would still buy the Powerlift 5 model ($94, varies by size and availability) as it is their most economical model. The difference in feel between the 1970s shoe and the current shoe is very small (I like consistency to keep technique efficient), they differ only in the feel of the now completely synthetic uppers. I have had other shoes Kail Sung (probably spelled wrong, it’s been a while, Russian shoe knock-offs, 1970s Korea), Kahru (1970s Finland), DooWin (1990s China), VS Athletics (2000s USA). VS still puts out an affordable and quality shoe ($89)(fun fact, both Kyle Pierce and I have worked with VS on a couple different shoe design projects in the past). If Puma made an affordable lifting shoe, I might get a pair and try them out, simply because I like their design aesthetics (and they share a common history with Adidas).
Powerlifting Shoes – For Back Squat and Bench Press I simply use my Adidas Weightlifting shoes. For the Squat, having squatted in nearly identical Adidas models ($94) for over 50 years it does not make sense, technically or economically, to have a different special Squat shoe. For the Bench Press, the sole of the Adidas shoe grabs the platform nicely and facilitates a solid set up on the bench. For the Deadlift I use a pair of VS Athletics throwing shoes ($54). The shoes are low, flat, and have a metatarsal strap to solidify foot anatomy and produce efficient force transfer over as short of a distance as possible. I also, very sporadically, compete in master’s shot put so having a single shoe that works perfectly for two sports avoids duplication and extra costs. I did at one time also Deadlift in knock-off Chuck E. Taylors ($23) but I sacrificed them by sawing one in half for an illustration I did for a book chapter on shoe construction relative to anatomy (and later in Barbell Sports). I would not do that to my ancient authentic Chuckies ($65).