Are these chemical reactions with amino acids possible, and if so, what are they called? In high school, I am taught that the synthesis of two amino acids of the form $\ce{H2NR1COOH}$ and $\ce{H2NR2COOH}$ is a condensation reaction that produces a polypeptide $$\ce{H2NR1COOH}+\ce{H2NR2COOH}\to\ce{H2NR1CONHR2COOH}+\ce{H2O}$$ where $\ce{R1}$ and $\ce{R2}$ are arbitrary components within each acid. Then I started experimenting with different positions that yield isomers of the polypeptide.
Examples

Water is formed though the pairing of the two $-\ce{OH}$ groups $$\ce{H2NR1COOH}+\ce{HOOCR2NH2}\to\ce{H2NR1COOCOR2NH2}+\ce{H2O}$$ which forms something that resembles an ester?

Water is formed through the pairing of the $-\ce{OH}$ and $\ce{H2N}-$ groups but the oxygen atom is taken instead from $\ce{C=O}$ $$\ce{H2NR1COOH}+\ce{H2NR2COOH}\to\ce{H2NR1C(OH)NR2COOH}+\ce{H2O}$$ where essentially the hydrogen atom from $-\ce{CONH}$ has shifted one place.



Are these reactions directly possible? If so, are there special names for
a) the reaction and
b) the type of product formed?

I am at an elementary level in chemistry so more detailed explanations would be appreciated.
Reaction 1

The product is called carboxylic acid anhydride. If you make sure the amino groups don't react (protecting and deprotecting) and get rid of water to prevent back-reactions, this reaction is possible.

Reaction 2

The product could be called a keto-enol tautomer (here, the enol form). The keto form is much more stable, so this product - even if made as an intermediate - would isomerize to the standard peptide bond.